representations, introduce the above grading on each of the summands, and
combine them into the grading of $V$. So, from now on we assume that $V$ is an
arbitrary (not necessarily irreducible) finite-dimensional representation of
$\mathfrak{g}$, equipped with the above grading. Let $\phi$ be the induced
filtration of $V$, which is by definition stabilized by $P$.

Let $G$ be a connected semisimple Lie group with a Lie algebra $\mathfrak{g}$,
and let $P$ be the parabolic subgroup corresponding to the subalgebra
$\mathfrak{p}$. The homogeneous flag variety $G/P$ is a rational homogeneous
variety. Assume that the representation $\mathfrak{g}\to\mathfrak{gl}(V)$ can
be extended to the representation $G\to GL(V)$ of the Lie group $G$. Let

$\varphi_{\text{model}}\colon G/P\to\operatorname{Flag}(V,\phi)$

be the standard embedding of type $(\mathfrak{g}_{-},V,G)$.

######  Lemma 2.

Suppose that $V$ is isotypic, that is all highest weights of $V$ coincide.
Then the relative prolongation $\operatorname{Prol}(\mathfrak{g}_{-},V)$ is
equal to $\mathfrak{g}+Z(\mathfrak{g})\subset\mathfrak{gl}(V)$, where
$Z(\mathfrak{g})$ is the centralizer of $\mathfrak{g}$ in $\mathfrak{gl}(V)$.

###### Proof.

It is sufficient to prove lemma in the complex case, as the real case
immediately follows via the complexification argument. Consider
$\mathfrak{g}$-module $\mathfrak{gl}(V)$. Assume first that $V$ is
irreducible. As $\operatorname{Prol}(\mathfrak{g}_{-},V)$ contains
$\mathfrak{g}$, it is a submodule of the $\mathfrak{g}$-module
$\mathfrak{gl}(V)$. Let $U$ be any graded submodule of
$\operatorname{Prol}(\mathfrak{g}_{-},V)$ such that $\mathfrak{g}\cap U=0$. By
definition $U$ is concentrated in the non-negative degree. Hence, it generates
a $\mathfrak{g}$-invariant subalgebra $\mathfrak{n}$ that is also concentrated
in a non-negative degree. In particular, $\mathfrak{g}+\mathfrak{n}$ is also a
subalgebra. As $\mathfrak{g}$-module $V$ is irreducible, so is
$(\mathfrak{g}+\mathfrak{n})$-module $V$. Hence, $\mathfrak{g}+\mathfrak{n}$
is reductive. In particular, the dimensions of subspaces of opposite degrees
should coincide. This is only possible if $\mathfrak{n}$, and thus $U$ is
concentrated in degree $0$. As $[\mathfrak{g}_{-},U]\subset U$, this
immediately implies that $[\mathfrak{g}_{-},U]=0$, and $U\subset
Z(\mathfrak{g})$.

Let $V$ be now an arbitrary isotypic $\mathfrak{g}$-module, that is $V$ is
isomorphic to $V^{\prime}\otimes\mathbb{C}^{k}$, where $V^{\prime}$ is
irreducible and $\mathbb{C}^{k}$ is a trivial $\mathfrak{g}$-module. We note
that $\mathfrak{g}$-module $\mathfrak{gl}(V)$ is isomorphic to
$\mathfrak{gl}(V^{\prime})\otimes\mathfrak{gl}(k,\mathbb{C})$ and any its
irreducible submodule has the form $U\otimes L$, where $U$ is an irreducible
submodule of $\mathfrak{gl}(V^{\prime})$ and $L$ is a $1$-dimensional subspace
in $\mathfrak{gl}(k,\mathbb{C})$. If such submodule lies in
$\operatorname{Prol}(\mathfrak{g}_{-},V)$ and has zero intersection with
$\mathfrak{g}$, then it is concentrated in the non-negative degree. It is easy
to see that $U$ itself lies in
$\operatorname{Prol}(\mathfrak{g}_{-},V^{\prime})$. By above this means that
$U\subset Z(\mathfrak{g})$. This completes the proof. ∎

######  Remark 2.

The condition of $V$ to be isotypic can not be dropped. The simplest counter-
example is given by $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$ and
$V=V_{1}\oplus V_{2}$, where $V_{1}$ and $V_{2}$ are two irreducible
representations of $\mathfrak{sl}(2,\mathbb{C})$ of different dimension.

In the following we always assume that the $\mathfrak{g}$-module $V$ is
isotypic. We also say that $\varphi_{\text{model}}$ is _a standard embedding_
of $G/P$ corresponding to the $\mathfrak{g}$-module $V$.

###  5.2. Harmonic analysis on semisimple Lie algebras

Let us introduce an invariant symmetric bilinear form on $\mathfrak{gl}(V)$ as
follows:

$\operatorname{Tr}(A,B)=\operatorname{tr}AB,\quad\text{ for all
}A,B\in\mathfrak{gl}(V).$

It is clear that the restriction of $\operatorname{Tr}$ to
$\mathfrak{gl}_{i}(V)\times\mathfrak{gl}_{j}(V)$ is identically zero for
$i+j\neq 0$ and is non-degenerate for $i+j=0$.

Note that the restriction of $\operatorname{Tr}$ to any semisimple subalgebra
of $\mathfrak{gl}(V)$ is non-degenerate. In particular, the restriction of
$\operatorname{Tr}$ to $\mathfrak{g}$ is non-degenerate, and we have
$\mathfrak{gl}(V)=\mathfrak{g}+\mathfrak{g}^{\perp}$, where
$\mathfrak{g}^{\perp}$ is an orthogonal complement of $\mathfrak{g}$ in
$\mathfrak{gl}(V)$ with respect to $\operatorname{Tr}$. Similarly, let
$\bar{\mathfrak{g}}$ be the prolongation of $(\mathfrak{g}_{-},V)$. Then
according to Lemma 2, we have
$\bar{\mathfrak{g}}=\mathfrak{g}+Z(\mathfrak{g})$.

######  Lemma 3.

We have the following orthogonal decomposition with respect to
$\operatorname{Tr}$:

$\mathfrak{gl}(V)=\mathfrak{g}\oplus
Z(\mathfrak{g})\oplus\bar{\mathfrak{g}}^{\perp}.$

###### Proof.

It is sufficient to consider only the complex case. As above, decompose $V$ as
$V^{\prime}\otimes\mathbb{C}^{k}$, where the module $V^{\prime}$ is
irreducible. Then from Schur’s lemma, we know that
$Z(\mathfrak{g})=\mathfrak{gl}(k,\mathbb{C})$. Thus, it is easy to see that
the restriction of $\operatorname{Tr}$ on $Z(\mathfrak{g})$ is non-degenerate.
Moreover, for any $X\in\mathfrak{g}$ and $A\in Z(\mathfrak{g}),$ we have
$\operatorname{tr}(XA)=\operatorname{tr}(X)\operatorname{tr}(A)=0$. ∎

Note that all three components of this decomposition are $\mathfrak{g}$\- and
$Z(\mathfrak{g})$-invariant. Moreover, since the Lie algebra $\mathfrak{g}$ is
graded and $Z(\mathfrak{g})\subset\mathfrak{gl}_{0}(V)$, we see that this
decomposition (namely, the subspace $\bar{\mathfrak{g}}^{\perp}$) is also
compatible with the grading of $\mathfrak{gl}(V)$.

Let $(C=\sum_{q}C^{q},\partial)$ be the cochain complex associated with the
$\mathfrak{g}_{-}$-module $\mathfrak{gl}(V)$. It inherits the grading from the
gradings of $\mathfrak{g}_{-}$ and $\mathfrak{gl}(V)$:

$C^{q}=\sum_{p}C_{p}^{q},\quad C_{p}^{q}=\\{c\in
C^{q}:c(\mathfrak{g}_{-i_{1}}\wedge\dots\wedge\mathfrak{g}_{-i_{q}})\subset\mathfrak{gl}_{p-i_{1}-\dots-
i_{q}}(V)\\}.$

It is clear that the coboundary operator $\partial$ preserves grading and,
hence, maps $C_{p}^{q}$ to $C_{p}^{q+1}$.

Similarly, for any $\mathfrak{g}_{-}$-invariant graded subspace $W$ in
$\mathfrak{gl}(V),$ denote by $C(W)=\sum C^{q}_{p}(W)$ the cochain complex
associated with the $\mathfrak{g}_{-}$-module $W$. We shall be particularly
interested in $C(\bar{\mathfrak{g}}^{\perp})$.

Any cochain $c\in C^{q}$ can be uniquely decomposed into the sum
$c=c_{I}+c_{II}$ according to the decomposition
$\mathfrak{gl}(V)=\bar{\mathfrak{g}}\oplus\bar{\mathfrak{g}}^{\perp}$.

The cohomology $H^{q}_{p}(\mathfrak{g}_{-},\bar{\mathfrak{g}}^{\perp})$ can be
effectively computed using Kostant theorem [15] and the notion of normality
formalized by Tanaka [35]. Namely, we can extend the scalar products on
$\mathfrak{g}$ and $V$ introduced in subsection 5.1 to the spaces $C^{q}$ and
introduce the adjoint coboundary operator $\partial^{*}\colon C^{q+1}\to
C^{q}$ and the Laplacian
$\Delta=\partial\partial^{*}+\partial^{*}\partial\colon C^{q}\to C^{q}$. The
standard argument shows that $\ker\Delta=\ker\partial\cap\ker\partial^{*}$ and
we have the direct product decomposition
$C^{q}=\operatorname{im}\partial\oplus\operatorname{im}\partial^{*}\oplus\ker
Delta$. We shall call an element $c\in C^{q}$ _harmonic_ , if $\Delta c=0$,
which is equivalent to $\partial c=\partial^{*}c=0$. Additionally, for any
$\chi\in C^{q}$ we shall call the _harmonic part of $\chi$_ and denote by
$H\chi$ the projection on $\chi$ to $\ker\Delta$ in the above direct product
decomposition.

###  5.3. Normal reductions

Let now $(M,\mathfrak{f})$ be an arbitrary filtered manifold of constant type
$\mathfrak{m}=\mathfrak{g}_{-}$. Let
$\varphi\colon(M,\mathfrak{f})\to\operatorname{Flag}(V,\phi)$ be arbitrary
osculating map. Define
$\bar{\mathfrak{g}}^{\prime}=\bar{\mathfrak{g}}^{\perp}$ in the condition (C1)
and $W=\phi^{1}\ker\partial^{*}$ in the condition (C2) of subsection 4.6.

In particular, an extrinsic Cartan connection associated with the embedding
$\varphi$ is $W$-normal, if and only if we have $\partial^{*}\chi=0$. We shall
call such extrinsic Cartan connections just normal.

Applying Theorem 1, we immediately get

######  Theorem 2.

For any embedding
$\varphi\colon(M,\mathfrak{f})\to\operatorname{Flag}(V,\phi)$ of type
$(\mathfrak{g}_{-},V)$, there is a unique normal extrinsic Cartan connection
$\omega\colon TP\to\mathfrak{gl}(V)$ on the principal
$\overline{G}^{0}$-bundle $P\to M$.

###  5.4. Vanishing of cohomology groups and rigidity

According to Corollary of Theorem 1, the structure function $\chi=\sum_{k\geq
1}\chi_{k}\colon$
$P\to\text{Hom}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})_{k}$ is a
_complete invariant_ of the embedding $\varphi$.

Using 5.2, we identify $H^{1}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})$
with $\ker\Delta=\ker\partial\cap\ker\partial^{*}$ and assume that the
harmonic part $H\chi$ of $\chi$ takes values in
$H^{1}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})$. It is the _fundamental
invariant_ of the embedding $\varphi$ in the following sense.

######  Proposition 9.

We have $\chi=0$ if and only if $H\chi=0$. And in this case the embedding
$\varphi$ is locally equivalent to the standard embedding of type
$(\mathfrak{g}_{-},V):$

$\varphi_{\text{model}}\colon G/G^{0}\to
L/L^{0}\subset\operatorname{Flag}(V,\phi).$

###### Proof.

Indeed, assume that for some $k\geq 0$ we have $\chi_{i}=0$ for all $i\leq k$.
Then equation (10) from the proof of Theorem 1 implies also that
$\partial\chi_{k+1}=0$. Since $\chi$ is normal, we also have
$\partial^{*}\chi_{k+1}=0$. So, $\chi_{k+1}=H\chi_{k+1}$. By induction, we get
that $\chi=0$ if and only if $H\chi=0$. ∎

In particular, if the cohomology spaces
$H^{1}_{r}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})$ vanish identically
for all $r\geq 1$, then any embedding $\varphi$ of type $(\mathfrak{g}_{-},V)$
is locally equivalent to the standard one. We say in this case that the
standard embedding $\varphi_{\text{model}}$ is _rigid_.

So, we arrive at a natural question, for which irreducible representations the
positive part of the cohomology
$H^{1}_{+}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})$ does not vanish. By
the following Proposition, the only rational homogeneous varieties of simple
Lie groups, which might be non-rigid are limited to (i) projective spaces,
(ii) quadrics, (iii) $A$-type (Lagrange) contact spaces, (iv) $C$-type
(projective) contact spaces.

######  Proposition 10.

Let $\mathfrak{g}=\bigoplus\mathfrak{g}_{p}$ be a simple graded Lie algebra
and $U$ an irreducible submodule of the $\mathfrak{g}$-module
$\mathfrak{gl}(V)$. The cohomology group $H^{1}_{r}(\mathfrak{g}_{-},U)$
vanishes for $r\geq 1$ except for the following cases$:$

  1. (1)

$(A_{3},\Sigma)$ with $\Sigma=\\{\alpha_{2}\\}$,
$(A_{l},\Sigma)\ (l\geq 1)$ with $\Sigma=\\{\alpha_{1}\\}$,
$\\{\alpha_{l}\\}$, or $\\{\alpha_{1},\alpha_{l}\\}$,

  2. (2)

$(B_{2},\Sigma)$ with $\Sigma=\\{\alpha_{1}\\}$, or $\\{\alpha_{2}\\}$,
$(B_{l},\Sigma)\ (l\geq 3)$ with $\Sigma=\\{\alpha_{1}\\}$,

  3. (3)

$(C_{l},\Sigma)\ (l\geq 3)$ with $\Sigma=\\{\alpha_{1}\\}$,

  4. (4)

$(D_{4},\Sigma)$ with $\Sigma=\\{\alpha_{1}\\}$, $\\{\alpha_{3}\\}$, or
$\\{\alpha_{4}\\}$,
$(D_{l},\Sigma)\ (l\geq 5)$ with $\Sigma=\\{\alpha_{1}\\}$,

where $\Sigma$ denotes the subset of the simple roots
$\\{\alpha_{1},\cdots,\alpha_{l}\\}$ which defines the gradation of
$\mathfrak{g}$, that is, $\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{p}$ for
$\alpha=a_{1}\alpha_{1}+\cdots+a_{l}\alpha_{l}$ if
$\sum_{\alpha_{i}\in\Sigma}a_{i}=p$.

###### Proof.

Let $\mu$ be the lowest weight of $U$. It is written as a linear combination
of the fundamental weights: $\mu=\sum_{i=1}^{l}\mu_{i}\omega_{i}$ with non-
positive integers $\mu_{i}$, where $\omega_{i}$ is the $i$-th fundamental
weight. By the theorem of Kostant [15], we have

$H^{1}(\mathfrak{g}_{-},U)=\bigoplus_{\alpha\in\Sigma}L^{\mathfrak{g}_{0}}(-\sigma_{\alpha}.(-\mu)),$

where $L^{\mathfrak{g}_{0}}(-\sigma_{\alpha}.(-\mu))$ denotes a
$\mathfrak{g}_{0}$-irreducible module of lowest weight:

$-\sigma_{\alpha}.(-\mu)=-(\sigma_{\alpha}(-\mu+\rho)-\rho)=\sigma_{\alpha}(\mu-\rho)+\rho,$

$\sigma_{\alpha}$ being the reflection with respect to $\alpha$ and
$\rho=\omega_{1}+\cdots+\omega_{l}$. Since
$\sigma_{\alpha_{i}}(\omega_{j})=\omega_{j}-\delta_{ij}\alpha_{i},$ we have

$\sigma_{\alpha_{i}}(\mu-\rho)+\rho=\sigma_{\alpha_{i}}(\mu)+\alpha_{i}.$

Now let us compute the degree of this weight. Since the grading of $U$ is
determined by the eigenvalues of the grading element $E\in\mathfrak{g}$, the
degree of $\lambda=\sum\lambda_{j}\alpha_{j}$ is defined as

$\deg(\lambda)=\sum_{\alpha_{i}\in\Sigma}\lambda_{i}.$

For $\alpha_{i}\in\Sigma$ we have

$\deg(\sigma_{\alpha_{i}}(\mu)+\alpha_{i})=\deg(\sigma_{\alpha_{i}}(\mu))+1.$

Let $C^{-1}=(p_{ij})$ be the inverse of the Cartan matrix $C$. Note that its
components $p_{ij}$ are all positive. We have

$\sigma_{\alpha_{i}}(\mu)=\sigma_{\alpha_{i}}\left(\sum_{j=1}^{l}\mu_{j}\omega_{j}\right)=\sum_{j=1}^{l}\mu_{j}\omega_{j}-\mu_{i}\alpha_{i}=\sum_{j,k=1}^{l}\mu_{j}p_{jk}\alpha_{k}-\mu_{i}\alpha_{i}.$

Therefore we have

$\displaystyle\deg(\sigma_{\alpha_{i}}(\mu))$
$\displaystyle=\sum_{j=1}^{l}\mu_{j}\sum_{k^{\prime}}p_{jk^{\prime}}-\mu_{i}$
$\displaystyle=\sum_{j\neq
i}\mu_{j}\sum_{k^{\prime}}p_{jk^{\prime}}+\mu_{i}\left(\sum_{k^{\prime}}p_{ik^{\prime}}-1\right),$

where the summation $\sum_{k^{\prime}}$ is taken over all $k^{\prime}$ such
that $\alpha_{k^{\prime}}\in\Sigma$. We then deduce from this that if
$(\mathfrak{g},\Sigma)$ has the following property:

$\sum_{\alpha_{k}\in\Sigma}p_{ik}-1>0\quad\text{for all }i\text{ such that
}\alpha_{i}\in\Sigma,$

then $H^{1}_{r}(\mathfrak{g}_{-},V)=0$ for $r\geq 1$. Now the theorem follows
immediately from the table of $(p_{ij})$ as found, for instance, in Humphreys’
book [10, p.69]. ∎

Consider now the cases from Proposition 10 in more detail. Let
$\mathfrak{p}\subset\mathfrak{g}$ be the parabolic subalgebra in the complex
simple Lie algebra that corresponds to one of the cases in Proposition 10.
Considering $\mathfrak{p}$ up to $\operatorname{Aut}(\mathfrak{g})$, is
equivalent to treating $\Sigma$ up to the automorphism of the Dynkin diagram
of the corresponding root system. Taking this into account, we may reduce the
number of cases to one of the following:

  1. (1)

$(A_{l},\\{\alpha_{1}\\})$, $l\geq 1$,

  2. (2)

$(B_{l},\\{\alpha_{1}\\})$, $l\geq 2$,

  3. (3)

$(C_{l},\\{\alpha_{1}\\})$, $l\geq 2$,

  4. (4)

$(D_{l},\\{\alpha_{1}\\})$, $l\geq 4$,

  5. (5)

$(A_{3},\\{\alpha_{2}\\})$,

  6. (6)

$(A_{l},\\{\alpha_{1},\alpha_{l}\\})$, $l\geq 2$.

We use here the identification of $(B_{2},\\{\alpha_{2}\\})$ with
$(C_{2},\\{\alpha_{1}\\})$.

In the case $(A_{l},\\{\alpha_{1}\\}),$ we get

$\displaystyle\deg(\sigma_{\alpha_{1}}(\mu))$
$\displaystyle=\sum_{j=2}^{l}p_{j1}\mu_{j}+\mu_{1}(p_{11}-1)$
$\displaystyle=\frac{1}{l+1}\left(\sum_{j=2}^{l}(l+1-j)\mu_{j}-\mu_{1}\right)\geq
0.$

This gives an explicit condition on the lowest weight $\mu$ of the module $U$,
such that $H^{1}(\mathfrak{g}_{-},U)$ is concentrated in the positive degree.

In $B_{l}$, $C_{l}$, and $D_{l}$ cases, we note that $p_{11}=1$. So, we get

$\deg(\sigma_{\alpha_{1}}(\mu))=\sum_{j=2}^{l}p_{j1}\mu_{j}+\mu_{1}(p_{11}-1)=\sum_{j=2}^{l}p_{j1}\mu_{j}\geq
0.$

Since $\mu_{j}\leq 0$, we immediately get that necessarily
$\mu_{2}=\dots=\mu_{l}=0$, and under this condition,
$H^{1}(\mathfrak{g}_{-1},U)$ is concentrated in degree $1$.

Similarly, in the case $(A_{3},\\{\alpha_{2}\\}),$ we have $p_{22}=1$. So, we
get that $H^{1}(\mathfrak{g}_{-1},U)$ is concentrated in positive degree if
any only if $\mu_{1}=\mu_{3}=0$, and this degree is equal to 1 in this case.

Finally, in the case $(A_{l},\\{\alpha_{1},\alpha_{l}\\}),$ we get

$\deg(\sigma_{\alpha_{1}}(\mu))=\sum_{j=2}^{l}(p_{j1}+p_{jl})\mu_{j}+\mu_{1}(p_{11}+p_{1l}-1)\\\
=\sum_{j=2}^{l}\mu_{j}\geq 0.$

So, again we get $\mu_{2}=\dots=\mu_{l}=0$ as the necessary condition for the
module $L^{\mathfrak{g}_{0}}(-\sigma_{\alpha_{1}}.(-\mu))\subset
H^{1}(\mathfrak{g}_{-},U)$ to be concentrated in the positive degree, which is
equal to 1 in this case.

Similarly, $L^{\mathfrak{g}_{0}}(-\sigma_{\alpha_{l}}.(-\mu))\subset
H^{1}(\mathfrak{g}_{-},U)$ is concentrated in positive degree (equal to $1$)
only for modules with the lowest weight $\mu_{l}\omega_{l}$, $\mu_{l}\leq 0$.

Summing up, we get

######  Proposition 11.

Let $\mu=\mu_{1}\omega_{1}+\dots+\mu_{l}\omega_{l}$, $\mu_{i}\leq 0$, be the
lowest weight of an irreducible module $U$ over one of the simple Lie algebras
of rank $l$ from Proposition 10. Suppose $H^{1}(\mathfrak{g}_{-},U)$ has non-
trivial components in positive degree. Then

  1. (1)

in the case $(A_{l},\\{\alpha_{1}\\}),$ we have

$N=\sum_{j=2}^{l}(l+1-j)\mu_{j}-\mu_{1}\geq 0,$

where $N$ is a multiple of $l+1$. Then $H^{1}(\mathfrak{g}_{-},U)$ is an
irreducible $\mathfrak{g}_{0}$-module concentrated in degree $N/(l+1)+1$;

  2. (2)

in the cases $B_{l}$ ($l\geq 2$), $C_{l}$ ($l\geq 2$), $D_{l}$ ($l\geq 4$) and
$\Sigma=\\{\alpha_{1}\\}$, and for $A_{3}$, $\Sigma=\\{\alpha_{2}\\}$, we have
$\mu_{i}=0$ unless $\alpha_{i}\in\Sigma$. Then $H^{1}(\mathfrak{g}_{-},U)$ is
an irreducible $\mathfrak{g}_{0}$-module concentrated in degree $1$;

  3. (3)

in the case $(A_{l},\\{\alpha_{1},\alpha_{l}\\})$, we have
$\mu_{2}=\dots=\mu_{l}=0$ or $\mu_{1}=\dots=\mu_{l-1}=0$. Each of these
conditions leads to an irreducible $\mathfrak{g}_{0}$-module in
$H^{1}(\mathfrak{g}_{-},U)$ concentrated in degree $1$.

All other cases from proposition 10 are equivalent to the above via the
automorphisms of the Dynkin diagram of the corresponding root system.

###  5.5. Simple Lie algebras of rank $2$ and their adjoint representations

In view of Propositions 10 and 11 let us consider, as first examples of
extrinsic parabolic geometry on non-trivial filtered manifolds, the following
simple Lie algebras of rank $2$ with contact type gradings and their adjoint
representations:

  * (i)

$(A_{2},\\{\alpha_{1},\alpha_{2}\\})$, $\mu=\omega_{1}+\omega_{2}$;

  * (ii)

$(C_{2},\\{\alpha_{1}\\}),$ $\mu=2\omega_{1}$;

  * (iii)

$(G_{2},\\{\alpha_{2}\\})$, $\mu=\omega_{2}$.

where $\\{\alpha_{i}\\}$ indicates the grading, and $\mu$ is the highest
weight of the adjoint representation.

Throughout this subsection $\mathfrak{g}$ ($=\oplus\mathfrak{g}_{p}$) will
denote one of the above simple Lie algebras with a contact type grading, that
is, the negative part $\mathfrak{g}_{-}$ of $\mathfrak{g}$ is a Heisenberg Lie
algebra, or more precisely,
$\mathfrak{g}_{-}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}$,
$\dim\mathfrak{g}_{-2}=1$, and the bracket
$\mathfrak{g}_{-1}\times\mathfrak{g}_{-1}\to\mathfrak{g}_{-2}$ is non-
degenerate. In the cases (i), (ii) $\dim\mathfrak{g}_{-}=3$ and $5$ in the
case (iii).

The representation $\rho\colon\mathfrak{g}\to\mathfrak{gl}(V)$ that we
consider here is the adjoint representation, so that $V=\mathfrak{g}$, while
we give a grading $\oplus V_{i}$ on $V$ by shifting:
$V_{i}=\mathfrak{g}_{i-2}$, to adjust to the usual orders of related
differential equations. We fix a filtration $\phi$ of $V$ by setting
$\phi^{i}=\oplus_{p\geq i}V_{p}$.

In each case we are going to inspect an osculating map

$\varphi\colon(M,\mathfrak{f})\to L/L^{0}\subset\operatorname{Flag}(V,\phi)$

and the associated differential equation of symbol type
$(\mathfrak{g}_{-},V,L)$ with a suitable choice of a Lie subgroup $L$ of
$GL(V)$. Note that the filtered manifold $(M,\mathfrak{f})$ has
$\mathfrak{g}_{-}$ as symbol, and therefore is a contact manifold of dimension
$3$ or $5$.

Case (i). In this case $\mathfrak{g}=\mathfrak{sl}(U)$ for a 3-dimensional
vector space $U$, equipped with the contact grading $\oplus\mathfrak{g}_{p}$,
which then satisfies

$\dim\mathfrak{g}_{\pm 2}=1,\dim\mathfrak{g}_{\pm 1}=\dim\mathfrak{g}_{0}=2.$

As already mentioned, $V=\mathfrak{g}$ and $V_{i}=\mathfrak{g}_{i-2}$.

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, say $SL(U)$. Then,
taking the $G$-orbit through $\phi$, we have the standard model embedding

$\varphi_{\mathrm{model}}\colon G/G^{0}\to
GL(V)/GL(V)^{0}=\operatorname{Flag}(V,\phi),$

where $G^{0}$ and $GL(V)^{0}$ are isotropy subgroups at $\phi$.

If $\varphi\colon(M,\mathfrak{f})\to
L/L^{0}\subset\operatorname{Flag}(V,\phi)$ is an osculating map of type
$(\mathfrak{g}_{-},V,L)$, then, since $V$ is generated by $V_{4}$ as
$\mathfrak{g}_{-}$ module, $\varphi$ is generated as an osculating map of the
projective embedding

$\varphi^{4}\colon(M,\mathfrak{f})\to P(V),$

which sends $x\in M$ to the 1-dimensional subspace $\varphi^{4}(x)$ of $V$.
Thus the extrinsic geometry of a 3-dimensional contact manifold in a
projective space is prolonged (or lifted) to that in a flag variety.

In particular, the standard model embedding $\varphi_{\mathrm{model}}$ may be
viewed as the prolongation of the standard projective embedding
$\varphi^{4}_{\mathrm{model}}$, which may be called the contact Segre
embedding: It is isomorphic to the projective embedding

$P(U)\breve{\times}P(U^{*})\to\ P(U\breve{\otimes}U^{*})$

induced from the tensor map $U\times U^{*}\ni(u,\alpha)\mapsto
u\otimes\alpha\in U\otimes U^{*}$, where $P(U)\breve{\times}P(U^{*})$ denotes
the set of all $([u],[\alpha])\in P(U)\times P(U^{*})$ such that $\langle
u,\alpha\rangle=0$, and $U\breve{\otimes}U^{*}$ the linear hull of all
$u\otimes\alpha$ such that $\langle u,\alpha\rangle=0$. Note that
$P(U)\breve{\times}P(U^{*})$ carries a $SL(U)$-invariant contact structure and
isomorphic to the projective cotangent bundle of $P(U)$ and that
$U\breve{\otimes}U^{*}$ can be identified with $\mathfrak{sl}(U)$.

As a choice of transformation group $L$ we may take $O(V)$ ($=O(V,\kappa)$),
where $\kappa$ is the Killing form of $\mathfrak{g}$. Since it is left
invariant with respect to the adjoint action of $G$, we have natural
embeddings:

$G\subset O(V)\subset
GL(V),\quad\mathfrak{g}\subset\mathfrak{o}(V)\subset\mathfrak{gl}(V).$

Note that the embeddings above of Lie algebras preserve the grading, where the
gradings of $\mathfrak{o}(V)$ and $\mathfrak{gl}(V)$ are these that are
induced from the filtration $\phi$ of $V$. Note that $O(V,\kappa)$ is
isomorphic to $O(5,3)$ in the real case.

As to the relative prolongation of $\mathfrak{g}_{-}$, by lemma 2 we have:

$\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{o}(V))=\mathfrak{g},\quad\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{gl}(V))=\bar{\mathfrak{g}}=\mathfrak{g}\oplus\mathfrak{z}.$

Now let us compute the cohomology groups
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{o}(V)/\mathfrak{g})$ and
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})$ which will
play a key role in the present extrinsic geometries. By the
Littlewood–Richardson rule we have the following direct sum decompositions
into irreducible $\mathfrak{g}$-modules:

$\mathfrak{g}=\Gamma_{1,1},\quad\bar{\mathfrak{g}}=\Gamma_{0,0}\oplus\Gamma_{1,1},\quad\mathfrak{l}=\Gamma_{1,1}\oplus\Gamma_{3,0}\oplus\Gamma_{0,3},$

and

$\mathfrak{gl}(V)=\Gamma_{0,0}\oplus\Gamma_{1,1}\oplus\Gamma_{1,1}\oplus\Gamma_{3,0}\oplus\Gamma_{0,3}\oplus\Gamma_{2,2},$

where $\Gamma_{\mu_{1},\mu_{2}}$ is a $\mathfrak{g}$-module with the highest
weight $\mu_{1}\omega_{1}+\mu_{2}\omega_{2}$.

Then, using Kostant’s theorem, we have:

$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})=H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{o}(V)/\mathfrak{g})=H^{1}_{1}(\mathfrak{g}_{-},\Gamma_{3,0})\oplus
H^{1}_{1}(\mathfrak{g}_{-},\Gamma_{0,3}),$

and the above $H^{1}_{1}$’s are irreducible $\mathfrak{g}_{0}$ modules of
highest weights $-\alpha_{1}+2\alpha_{2}$ and $2\alpha_{1}-\alpha_{2}$
respectively, hence both are 1-dimensional. This algebraic result suggests
that the extrinsic geometry with respect to $GL(V)$ reduces to that with
respect to the group $O(V)$ and the latter may have non-trivial geometric
invariants.

Let us now consider the differential equation $\mathcal{R}$ corresponding to
an osculating map $\varphi\colon(M,\mathfrak{f})\to
L/L^{0}\subset\operatorname{Flag}(V,\phi)$ of symbol type
$(\mathfrak{g}_{-},V,L)$. Here $L$ may be any subgroup of $GL(V)$ containing
$G$, but we regard $L$ as just $GL(V)$ for simplicity.

Let us first observe what is the standard model equation
$\mathcal{R}_{\mathrm{model}}$ defined on $G/G^{0}$ that corresponds to the
standard model embedding $\varphi_{\mathrm{model}}$.

We then find that the model equation can be locally expressed in the following
form of a system of (weighted) second order differential equations:

$\left\\{\begin{aligned} X^{2}u&=0,\\\ Y^{2}u&=0,\end{aligned}\right.$

with

$X=\frac{\partial}{\partial x}+\frac{1}{2}y\frac{\partial}{\partial z},\quad
Y=\frac{\partial}{\partial y}-\frac{1}{2}x\frac{\partial}{\partial z}.$

for a contact coordinate system $(x,y,z)$ of $G/G^{0}$: The contact
distribution $D$ is defined by a contact form

$\theta=dz-\frac{1}{2}(ydx-xdy),$

therefore $X,Y$ are sections of $D$.

We have already given several approaches to get the differential equation
corresponding to an embedding $\varphi$ to a flag variety (see Section 3 and
[23]). We leave to the reader to give (two) different proofs for this model
equation.

Now the differential equation corresponding to an arbitrary embedding
$\varphi\colon(M,\mathfrak{f})\to\operatorname{Flag}(V,\phi)$ of symbol
$(\mathfrak{g}_{-},V)$ can be written as

$\left\\{\begin{aligned} X^{2}u&=a_{1}Xu+a_{2}Yu+a_{3}u,\\\
Y^{2}u&=b_{1}Xu+b_{2}Yu+b_{3}u,\end{aligned}\right.$

where $a_{i},b_{i}$ are local functions on $M$ satisfying all the
compatibility conditions, and $X,Y$ are sections of the contact distribution
$D$ ($=\mathfrak{f}^{-1}$), and $u$ is a section of
$\mathfrak{f}^{-2}/\mathfrak{f}^{-1}$ viewed as a function on $M$ by taking a
local trivialization of the bundle, say $u=\langle\theta,\zeta\rangle$ for the
unknown section $\zeta$ of $\mathfrak{f}^{-2}/\mathfrak{f}^{-1}$.

We remark that the the space $\mathfrak{g}_{-1}$ has the
$\mathfrak{g}_{0}$-invariant splitting
$\mathfrak{g}_{-1}=\mathfrak{g}_{-1}^{L}\oplus\mathfrak{g}_{-1}^{R}$, where
$\dim\mathfrak{g}_{-1}^{L}=\dim\mathfrak{g}_{-1}^{R}=1$. This gives rise to a
splitting $D=E\oplus F$ of the contact distribution $D$, which is an extra
geometric structure on $M$ induced from the extrinsic geometry of embedding
$\varphi$. In the above equation we take $X$, $Y$ as sections of $E$, $F$
respectively.

In general, it is not easy to write down all the compatibility conditions. But
now consider the following equation of special form:

$\left\\{\begin{aligned} X^{2}u&=aYu,\\\ Y^{2}u&=0,\end{aligned}\right.$

where $a$ is an arbitrary constant. By computer aided computation, we find
that it still has an $8$-dimensional solution space spanned by

$\left\\{\begin{aligned} 1,&\ x,z+\frac{1}{2}xy,\\\ y&+\frac{a}{2}x^{2},\\\
-z&+\frac{1}{2}xy+\frac{a}{6}x^{3},\\\
x&(-z+\frac{1}{2}xy)+\frac{a}{12}x^{4},\\\
y&(z+\frac{1}{2}xy)+\frac{a}{2}x^{2}(z-\frac{1}{6}xy)-\frac{a^{2}}{60}x^{5},\\\
(z&+\frac{1}{2}xy)(-z+\frac{1}{2}xy)+\frac{a}{6}x^{3}z-\frac{a^{2}}{360}x^{6},\end{aligned}\right.$

This implies that the deformed equation satisfies all the compatibility
conditions and its solution will possibly give an explicit non-trivial (local)
deformation of the standard embedding. Actually to affirm this, see
Proposition 12 in the discussion of the next case (ii) .

This case can be further generalized to the case of
$(A_{2},\\{\alpha_{1},\alpha_{2}\\})$, $\mu=k\omega_{1}+l\omega_{2}$,
corresponding to the linear differential equation $X^{k+1}u=0$, $Y^{l+1}u=0$
and its deformation.

Even staying still in the adjoint representation of $\mathfrak{sl}(3)$, the
extrinsic geometry and the associated differential equations are very rich and
interesting, related to the nice symmetry of the star of Davide:
$\Gamma_{3,0}\oplus\Gamma_{0,3}$. We shall develop a detailed study of this
subject in the forthcoming paper.

Case (ii). We now consider the case where $\mathfrak{g}=\mathfrak{sp}(U)$,
where $U$ is a 4-dimensional vector space equipped with a non-degenerate
symplectic form. As in case (i), we endow $\mathfrak{g}$ with a contact type
grading so that

$\dim\mathfrak{g}_{\pm 2}=1,\dim\mathfrak{g}_{\pm
1}=2,\dim\mathfrak{g}_{0}=4.$

As already said, $V=\mathfrak{g}$ with $V_{i}=\mathfrak{g}_{i-2}$ as a graded
$\mathfrak{g}$ module. We assume also that $G=Sp(U)$.

Consider an osculating map

$\varphi\colon(M,\mathfrak{f})\to L/L^{0}\subset\operatorname{Flag}(V,\phi)$

for a Lie subgroup $L$ of $GL(V)$ containing $G$. Then it is generated by the
projective embedding

$\varphi^{4}\colon(M,\mathfrak{f})\to P(V).$

In particular the standard model embedding

$\varphi_{\mathrm{model}}\colon G/G^{0}\to\operatorname{Flag}(V,\phi)$

is generated by the projective embedding

$\varphi^{4}_{\mathrm{model}}\colon G/G^{0}\to P(V),$

which may be called the contact Veronese embedding: It is isomorphic to the
following projective embedding

$P(U)\to P(S^{2}(U)).$

Recall here that $\mathfrak{sp}(U)$ may be identified with $S^{2}(U)$.

As in case (i), as a choice of transformation group $L$ we can take $O(V)$
($=O(V,\kappa)$), where $\kappa$ is the Killing form of $\mathfrak{g}$. So, we
have natural embeddings:

$G\subset O(V)\subset
GL(V),\quad\mathfrak{g}\subset\mathfrak{o}(V,\kappa)\subset\mathfrak{gl}(V).$

Note that in the real case we have $\mathfrak{o}(V,\kappa)=\mathfrak{o}(6,4)$.

Similar to above, the relative prolongations of $\mathfrak{g}_{-}$ in
$\mathfrak{o}(V)$ and $\mathfrak{gl}(V)$ coincide with $\mathfrak{g}$ and
$\bar{\mathfrak{g}}=\mathfrak{g}+\mathfrak{z}$ respectively.

Let us compute cohomology groups
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{o}(V)/\mathfrak{g})$ and
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})$. In terms of
$\mathfrak{g}$-modules, we have the irreducible decomposition:

$\mathfrak{gl}(V)=\mathfrak{gl}(10)=\Gamma_{0,0}\oplus\Gamma_{0,1}\oplus\Gamma_{2,0}\oplus\Gamma_{0,2}\oplus\Gamma_{2,1}\oplus\Gamma_{4,0}.$

Here $\mathfrak{l}=\mathfrak{o}(6,4)=\Gamma_{2,0}\oplus\Gamma_{2,1}$. Since
the relative prolongation
$\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{o}(6,4))=\mathfrak{g}$,

$\bar{\mathfrak{g}}=\mathfrak{g}=\Gamma_{2,0}=\mathfrak{sp}(4),\quad\bar{\mathfrak{g}}^{\perp}=\Gamma_{2,1}.$

Then by Proposition 11, we have no fundamental invariants of $\varphi$:

$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{g}^{\perp})=0.$

On the other hand, if we take $\mathfrak{l}$ to be the whole
$\mathfrak{gl}(10)$, the relative prolongation is
$\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{gl}(10))=\mathfrak{g}\oplus\mathfrak{z}=\bar{\mathfrak{g}}$.
Then

$H^{1}_{+}(\mathfrak{g}_{-},\bar{\mathfrak{g}}^{\perp})=H^{1}_{1}(\mathfrak{g}_{-},\Gamma_{4,0})=S^{5}\mathbb{R}^{2}\quad(\dim=6).$

It corresponds to the highest weight $-\alpha_{1}+2\alpha_{2}$ (and the lowest
weight $\alpha_{1}-2\alpha_{2}$).

The model equation can be written in this case as:

$\left\\{\begin{aligned} X^{3}u&=0,\\\ XYXu&=0,\\\ YXYu&=0,\\\
Y^{3}u&=0.\end{aligned}\right.$

Now consider its deformed version:

($\mathcal{E}_{a}$) $\begin{cases}X^{3}u=aY^{2}u,\\\ XYXu=0,\\\ YXYu=0,\\\
Y^{3}u=0,\end{cases}$

where $a$ is an arbitrary constant. We can verify that it still has a
$10$-dimensional solution space spanned by

$\left\\{\begin{aligned} 1,&x,y,z,x^{2},xy,xz,\\\ y^{2}&+\frac{a}{3}x^{3},\\\
yz&-\frac{a}{24}x^{4},\\\ z^{2}&+\frac{a}{120}x^{5},\end{aligned}\right.$

which defines an explicit (local) deformation of the standard embedding.

Let us prove that this deformation is non-trivial.

######  Proposition 12.

If $a\neq 0$, then ($\mathcal{E}_{a}$) is not locally equivalent to
($\mathcal{E}_{0}$) as linear differential equations.

###### Proof.

Let us show that the structure function $\chi=\chi_{1}$ of an extrinsic bundle
$Q$ associated to the equation ($\mathcal{E}_{a}$) does not vanish if $a\neq
0$, which will then prove by Theorem 1 that ($\mathcal{E}_{a}$) is not
isomorphic to ($\mathcal{E}_{0}$) if $a\neq 0$.

We first note that since the principal parts of ($\mathcal{E}_{a}$) are the
same for all $a$, the symbol of ($\mathcal{E}_{a}$) is isomorphic to that of
the model equation and is equal to $(\mathfrak{g}_{-},V)$ as graded module,
where $\mathfrak{g}=\mathfrak{sp}(4,\mathbb{R})$ and $V=\mathfrak{g}$ as
graded module (modulo the shift of the grading).

Next, let us note that ($\mathcal{E}_{a}$) satisfies all compatibility
conditions for any $a$, as suggested by the computer computation mentioned
above.

Let $X,Y$ be as above and let $Z=-[X,Y]=\frac{\partial}{\partial z}$. By
simple computation, we have:

$\begin{cases}X^{2}Y=-XZ,\\\ Y^{2}X=-YZ,\end{cases}$

and for the 4-th order:

$\begin{cases}X^{4}&=-aYZ,\\\ X^{3}Y&=0,\\\ X^{2}Y^{2}&=\frac{1}{2}Z^{2},\\\
XY^{3}&=0,\\\ Y^{4}&=0,\\\ X^{2}Z&=0,\\\ XYZ&=-\frac{1}{2}Z^{2},\\\
Y^{2}Z&=0.\end{cases}$

The value of $Z^{2}$ remains free.

For the 5-th order we have:

$X^{5}=\frac{a}{2}Z^{2},$

and all other 5-th order derivatives:

$\displaystyle X^{4}Y,X^{3}Y^{2},\dots$ $\displaystyle X^{3}Z,X^{2}YZ,\dots$
$\displaystyle XZ^{2},YZ^{2}$

vanish.

We then see easily that all derivatives of order higher than $5$ vanish. Thus,
the equation ($\mathcal{E}_{a}$) is prolonged to a weighted involutive system
of order $5$ and has a 10-dimensional solution space.

Now note that since the symbol of ($\mathcal{E}_{a}$) is isomorphic to
$(\mathfrak{g}_{-},\operatorname{gr}V)$, this means that

$(\operatorname{gr}V)^{*}\cong
U(\mathfrak{g}_{-})\otimes(\operatorname{gr}_{-2}V)^{*}/I,$

where $U(\mathfrak{g}_{-})$ denotes the universal enveloping algebra of
$\mathfrak{g}_{-}$ and $I$ denotes the left $U(\mathfrak{g}_{-})$-module
generated by the generators of model equation ($\mathcal{E}_{0}$), that is

$X^{3}\otimes Z^{*},Y^{3}\otimes Z^{*},XYX\otimes Z^{*},YXY\otimes Z^{*},$

where $Z^{*}$ denotes the basis of $(\operatorname{gr}_{-2}V)^{*}$ dual to
$Z$.

Recalling our convention of grading, e.g.,
$\operatorname{gr}_{q}V^{*}=(\operatorname{gr}_{-q}V)^{*}$, we note that the
isomorphism above is an isomorphism of $\mathfrak{g}_{-}$-modules, which can
be given explicitly as follows.

Let $\\{A_{1},\dots,A_{10}\\}$ be a basis of $\mathfrak{sp}(4,\mathbb{R})$ and
$\\{A_{1}^{*},\dots,A_{10}^{*}\\}$ its dual basis of
$\mathfrak{sp}(4,\mathbb{R})^{*}$ given by

$\displaystyle A_{1}$ $\displaystyle=E_{14}$ $\displaystyle A_{4}$
$\displaystyle=E_{23}$ $\displaystyle A_{8}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{21}-E_{43})$ $\displaystyle A_{2}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{13}+E_{24})$ $\displaystyle A_{5}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{11}-E_{44})$ $\displaystyle A_{9}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{31}+E_{42})$ $\displaystyle A_{3}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{12}-E_{34})$ $\displaystyle A_{6}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(E_{22}-E_{33})$ $\displaystyle A_{10}$
$\displaystyle=E_{41}$ $\displaystyle A_{7}$ $\displaystyle=E_{32}.$

We identify $X=A_{8}$, $Y=A_{9}$, $Z=A_{10}$.

Then we can easily verify that the following map gives a
$\mathfrak{g}_{-}$-module isomorphism:

$\begin{pmatrix}A_{1}^{*}\\\ A_{2}^{*}\\\ \vdots\\\
A_{10}^{*}\end{pmatrix}\mapsto\begin{pmatrix}-\tfrac{1}{2}Z^{2}\\\ YZ\\\ XZ\\\
-Y^{2}\\\ -\tfrac{1}{\sqrt{2}}Z\\\ -\tfrac{1}{\sqrt{2}}(XY+YX)\\\ X^{2}\\\
Y\\\ X\\\ 1\end{pmatrix}\cdot Z^{*}.$

We denote by $P$ the 10 by 1 matrix of differential operators that appear in
the the right hand side of the above correspondence. Let
$\\{\theta_{1},\dots,\theta_{10}\\}$ be a fundamental system of solutions of
($\mathcal{E}_{a}$) and let

$\theta=(\theta_{1},\dots,\theta_{10}),\quad\Theta=P\theta,\quad\Phi=\Theta^{-1}.$

Note that $\Theta$ and $\Phi$ are $GL(V)\,(=GL(10,\mathbb{R})$)-valued local
functions on $M$. Similarly as in Example 4 we may regard $\Phi$ as giving a
cross-section:

$\begin{CD}Q(0)@=Q(0)@>{\iota}>{}>GL(V)\\\
@A{\Phi}A{}A@A{}A{\Theta^{-1}}A@V{}V{}V\\\
M@=M@>{\varphi}>{}>\operatorname{Flag}(V,\phi),\end{CD}$

where $\varphi$ is an embedding corresponding to ($\mathcal{E}_{a}$) and
$Q(0)$ the associated extrinsic bundle (recall Subsection 4.3).

Therefore, the $\mathfrak{gl}(V)$-valued form $\Omega$ given by

$\Omega=\Phi^{-1}\,d\Phi=-d\Theta\,\Theta^{-1}$

is the pull-back to $M$ of the canonical form $\omega(0)$ on $Q(0)$. Write

$\Omega=F_{0}\omega_{0}+F_{1}\omega_{1}+F_{2}\omega_{2},$

where $\\{\omega_{0},\omega_{1},\omega_{2}\\}$ is the coframe dual to
$\\{Z,X,Y\\}$. By using the relations of ($\mathcal{E}_{a}$) we can easily
write down $\\{F_{i}\\}$ and see that $F_{0}$ and $F_{2}$ do not depend on the
parameter $a$ and

$F_{1}=aE_{74}+\overset{\circ}{F}_{1},$

where $E_{74}$ is the matrix element of $(7,4)$ and $\overset{\circ}{F}_{1}$
does not depend on $a$.

Recalling the discussion of Section 4, we see that the 1-st order component
$\chi(0)_{1}$ of the structure function $\chi(0)$ of $Q(0)$ is then
represented by $\gamma=E_{74}\otimes A^{*}_{8}$.

Now take a closer look at the complex:

$0\to\bar{\mathfrak{g}}_{1}\xrightarrow{\partial}\operatorname{Hom}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})_{1}\to\xrightarrow{\partial}\operatorname{Hom}(\wedge^{2}\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})_{1}.$

If we write down explicitly the adjoint representation of
$\mathfrak{sp}(4,\mathbb{R})$ with respect to the basis
$\\{A_{1},A_{2},\dots,A_{10}\\}$, it is not difficult to verify that $\gamma$
is a cocycle but not a coboundary. Hence it defines a non-trivial cohomology
in $H^{1}_{1}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})$. It then
proves, by virtue of Corollary to Theorem 1, that ($\mathcal{E}_{a}$) is not
isomorphic to ($\mathcal{E}_{0}$) for any $a\neq 0$. ∎

It should be remarked that $\gamma=E_{74}\otimes A_{8}^{*}$ has weight
$\alpha_{1}-2\alpha_{2}$ and in turn gives a lowest weight vector of
$H^{1}(\mathfrak{g}_{-},\mathfrak{gl}(V)/\bar{\mathfrak{g}})$. Note also that
systems ($\mathcal{E}_{a_{1}}$) and ($\mathcal{E}_{a_{2}}$) for different non-
vanishing $a_{1}$ and $a_{2}$ are equivalent to each other.

Case (iii). Let us consider the adjoint representation
$\rho\colon\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$, where $\mathfrak{g}$
is the split simple Lie algebra of type $G_{2}$. For contact grading we have

$\mathfrak{g}=\sum_{i=-2}^{2}\mathfrak{g}_{i},\qquad\dim\mathfrak{g}_{\pm
2}=1,\dim\mathfrak{g}_{\pm 1}=\dim\mathfrak{g}_{0}=4.$

As before, $V=\mathfrak{g}$ and $V_{i}=\mathfrak{g}_{i-2}$.

Let $U$ be the $7$-dimensional standard representation of $\mathfrak{g}$. It
is easy to check that the $\mathfrak{g}$-module $\Lambda^{2}U$ decomposes as
$V\oplus U$. The projection to the first summand gives us
$\mathfrak{g}$-invariant map $\Lambda^{2}U\to V$, while the projection to the
second summand defines an anticommutative multiplication on $U$.

The standard embedding $\varphi_{\mathrm{model}}$ is generated in this case by
the following projective embedding:

$\varphi^{4}_{\mathrm{model}}\colon
G/G^{0}=N_{0}\operatorname{Gr}_{2}(U)\longrightarrow P^{13}=P(V),$

where $N_{0}\operatorname{Gr}_{2}(U)$ is the Grassmann manifold of all
$2$-dimensional null subalgebras in $U$. We may call this map the contact sub-
Plücker embedding.

As in the previous cases, we have

$\mathfrak{g}\subset\mathfrak{o}(V,\kappa)=\mathfrak{o}(8,6)\subset\mathfrak{gl}(V),$

where $\kappa$ is the Killing form of $\mathfrak{g}$. As in the previous
cases, we can choose $L=O(V,\kappa)=O(8,6)$ and
$\mathfrak{l}=\mathfrak{o}(V,\kappa)=\mathfrak{o}(8,6)$.

Note that $\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})=\mathfrak{g}$,
and we have the following decomposition of $\mathfrak{g}$-modules:

$\mathfrak{gl}(V)\supset\mathfrak{o}(8,6)=\Gamma_{0,1}\oplus\Gamma_{0,2},\quad\mathfrak{g}=\Gamma_{0,1},\
\mathfrak{g}^{\perp}=\Gamma_{0,2}.$

Then by Proposition 11, we get the fundamental invariants of $\varphi$:

$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{g}^{\perp})=H^{1}_{+}(\mathfrak{g}_{-},\Gamma_{0,2})=0.$

Similarly, if we take $\mathfrak{l}=\mathfrak{gl}(V)$, then
$\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})=\bar{\mathfrak{g}}=\mathfrak{g}+\mathfrak{z}$,
and

$\bar{\mathfrak{g}}^{\perp}=\Gamma_{0,2}\oplus\Gamma_{2,0}\oplus\Gamma_{3,0}.$

According to Proposition 11 we still have trivial cohomology describing the
fundamental invariants of osculating maps $\varphi\colon(M,\mathfrak{f})\to
L/L^{0}\subset\operatorname{Flag}(V,\phi)$. So the standard model embedding
$\varphi_{\mathrm{model}}$ is rigid for $L=L=O(8,6)$ as well as for $GL(V)$.

We remark that the first cohomology $H^{1}(\mathfrak{g}_{-},V)=\bigoplus
H^{1}_{r}(\mathfrak{g}_{-},V)$ represents a system of differential equations,
and $V=\oplus V_{q}$ represents its solution space, see [23]. In our case the
computation shows that

$H^{1}(\mathfrak{g}_{-},V)=H^{1}_{0}(\mathfrak{g}_{-},V)\cong\operatorname{Hom}_{0}(S^{2}\mathfrak{g}_{-1},V_{0}).$

Here $\operatorname{Hom}_{0}(S^{2}\mathfrak{g}_{-1},V_{0})$ is the
7-dimensional irreducible $\mathfrak{g}_{0}$-submodule of
$\operatorname{Hom}(S^{2}\mathfrak{g}_{-1},V_{0})$. Explicitly, we get the
system of (weighted) linear differential equations on $M$ which is
$G_{2}$-invariant and has $14$-dimensional solution space defining the
standard embedding:

$\left\\{\begin{aligned} Z^{2}u&=0,\\\ W^{2}u&=0,\\\ YZu&=0,\\\ XWu&=0,\\\
(2ZX+Y^{2})u&=0,\\\ (2YW+X^{2})u&=0,\\\
(XY+YX+WZ+ZW)u&=0,\end{aligned}\right.$

where we take the local coordinates $(x,y,z,w,v)$ in $X$ such that the contact
distribution is represented by $D=\\{dv-2ydx+xdy+zdw=0\\}$,
$Z=\frac{\partial}{\partial z},\ W=\frac{\partial}{\partial
w}-z\frac{\partial}{\partial v},\ X=\frac{\partial}{\partial
x}+2y\frac{\partial}{\partial v},\ Y=\frac{\partial}{\partial
y}-x\frac{\partial}{\partial v}$. See [12] for more details. Here $u$ is an
unknown function $u=u(x,y,z,w,v)$ on $X$.

The basis of the corresponding $14$-dimensional solution space is

$\left\\{\begin{aligned} {}&1,&&\deg:0\\\ {}&x,\ y,\ z,\ w,&&\deg:1\\\
{}&x^{2}-yw,\ y^{2}-xz,\ xy-zw-v,\ v,&&\deg:2\\\ {}&w(y^{2}-xz)-xv,\
x(y^{2}-xz)-yv,\\\ {}&w(xy-zw-v)-x(x^{2}-yw),\ x(xy-
zw-v)-y(x^{2}-yw),\quad&&\deg:3\\\ {}&v(xy-
zw-v)-(x^{2}-yw)(y^{2}-xz),&&\deg:4,\end{aligned}\right.$

which gives the coordinate representation of $\varphi_{\text{model}}^{4}$.

###  5.6. Realizability of deformations

Proposition 10 lists all possible types of non-rigid rational homogeneous
varieties of simple Lie groups. However, it does not provide explicit examples
when they are really rigid.

For quadrics one can take the standard representation of $B_{l}$, $D_{l}$ or
$A_{3}\cong D_{3}$, as a non-degenerate quadric in $P^{n}$ is clearly non-
rigid.

For projective spaces one can consider, for example, the osculating map
$\varphi\colon P^{n}\to\operatorname{Flag}(\mathbb{R}^{(n+1)(n+2)/2},\phi)$
generated by the Veronese embedding $\nu_{2}\colon P^{n}\to
P^{(n+1)(n+2)/2-1}$, which corresponds to the $S^{2}(V)$ representation of
$SL(V)$. The filtration $\phi$ has type $((n+1)(n+2)/2,n+1,1)$ in this case.

Let $(x_{1},\dots,x_{n})$ be affine coordinates on $P^{n}$. In these
coordinates the Veronese embedding is specified by the space of all
polynomials of degree $\leq 2$ in $x_{1},\dots,x_{n}$. The corresponding
$PSL(n+1,\mathbb{R})$ invariant system of PDEs:

$\frac{\partial^{3}u}{\partial x_{i}\partial x_{j}\partial x_{k}}=0.$

We can provide an explicit deformation of the Veronese embedding by replacing
the equation $\frac{\partial^{3}u}{(\partial x_{1})^{3}}=0$ with:

$\frac{\partial^{3}u}{(\partial x_{1})^{3}}=\frac{\partial^{2}u}{(\partial
x_{2})^{2}}.$

It is easy to see that this deformed system is still compatible and has the
solution space of dimension $(n+1)(n+2)/2$, thus providing an explicit
deformation of the Veronese embedding. Computer algebra calculations show that
the dimension of the symmetry algebra of this system is less than the one of
the trivial system implying that this deformation is non-trivial.

Consider now the adjoint variety of $\mathfrak{sp}(2n,\mathbb{R})$. It was
explicitly demonstrated in the previous subsection that it is non-rigid for
for the case of $\mathfrak{sp}(4,\mathbb{R})$.

Assume now that $n$ is arbitrary. Let $V=\mathfrak{sp}(2n,\mathbb{R})$ be the
adjoint representation. Its highest weight is $2\omega_{1}$, and
$\hat{\mathfrak{g}}=\mathfrak{csp}(2n,\mathbb{R})$. We have the following
decompositions of $\mathfrak{gl}(V)$ and
$\hat{\mathfrak{g}}^{\perp}\subset\mathfrak{gl}(V)$:

$\displaystyle\mathfrak{gl}(V)$
$\displaystyle=V_{4\omega_{1}}+V_{2\omega_{1}+\omega_{2}}+V_{2\omega_{2}}+V_{2\omega_{1}}+V_{\omega_{2}}+V_{0};$
$\displaystyle\hat{\mathfrak{g}}^{\perp}$
$\displaystyle=V_{4\omega_{1}}+V_{2\omega_{1}+\omega_{2}}+V_{2\omega_{2}}+V_{\omega_{2}}.$

In particular, $H^{1}_{+}(\mathfrak{g}_{-},V_{4\omega_{1}})$ does not vanish,
which suggests that the adjoint variety of $\mathfrak{sp}(2n,\mathbb{R})$ is
non-rigid for any $n\geq 3$.

Consider now $A$-type contact case in more detail. It corresponds to
$(A_{l},\\{\alpha_{1},\alpha_{l}\\})$. Its minimal embedding is given by the
highest root orbit of the adjoint representation of $SL(l+1,\mathbb{R})$.
However, it can be shown that this orbit is non-rigid only for $l=2$. Indeed,
we have the following decomposition of
$\hat{\mathfrak{g}}^{\perp}\subset\mathfrak{gl}(V)$,
$V=\mathfrak{sl}(l+1,\mathbb{R})$ depending on $l$:

$\displaystyle\hat{\mathfrak{g}}^{\perp}$
$\displaystyle=V_{2\omega_{1}+2\omega_{l}}+V_{2\omega_{1}+\omega_{l-1}}+V_{\omega_{2}+2\omega_{l}}+V_{\omega_{2}+\omega_{l-1}}+V_{\omega_{1}+\omega_{l}},$
$\displaystyle\quad l\geq 4;$ $\displaystyle\hat{\mathfrak{g}}^{\perp}$
$\displaystyle=V_{2\omega_{1}+2\omega_{3}}+V_{2\omega_{1}+\omega_{2}}+V_{\omega_{2}+2\omega_{3}}+V_{2\omega_{2}}+V_{\omega_{1}+\omega_{3}},$
$\displaystyle\quad l=3;$ $\displaystyle\hat{\mathfrak{g}}^{\perp}$
$\displaystyle=V_{2\omega_{1}+2\omega_{2}}+V_{3\omega_{1}}+V_{3\omega_{2}}+V_{\omega_{1}+\omega_{2}},$
$\displaystyle\quad l=2.$

In particular, we see that for $l\geq 3$ this decomposition has no components
$U$, for which the cohomology $H^{1}_{+}(\mathfrak{g}_{-},U)$ is non-trivial
(see Proposition 11).

######  Lemma 4.

Consider the injective map $\mathfrak{sl}(l+1,\mathbb{R})\to\mathfrak{gl}(V)$,
$l\geq 2$, where $V$ is the irreducible representation with the highest weight
$\rho=\omega_{1}+\omega_{2}+\dots+\omega_{l}$. Then the decomposition of
$\mathfrak{gl}(V)$ contains the components $V_{(l+1)\omega_{1}}$ and
$V_{(l+1)\omega_{l}}$ with multiplicity $1$.

###### Proof.

Indeed, the multiplicity of $V_{(l+1)\omega_{1}}$ and $V_{(l+1)\omega_{l}}$ in
the decomposition of $\mathfrak{gl}(V_{\rho})$ can be computed using the
Littlewood–Richardson rule [8]. To do this, we note that the representation
$V_{\rho}$ is self-dual, and, hence $\mathfrak{gl}(V_{\rho})$ is isomorphic to
$V_{\rho}\otimes V_{\rho}$. In terms of Young tableaux the representation
$V_{\rho}$ corresponds to the tableau $\lambda$ of shape $(l,l-1,\dots,1)$
($l$ boxes in the first row, $l-1$ boxes in the second row, …, $1$ box in the
last row).

As $V_{(l+1)\omega_{1}}$ and $V_{(l+1)\omega_{l}}$ are dual to each other, it
is clear that their multiplicities in the decomposition of $V_{\rho}\otimes
V_{\rho}$ are the same. So, we can concentrate on computing the multiplicity
of $V_{(l+1)\omega_{1}}$, which corresponds to any tableau of shape
$(k+l+1,k,\dots,k)$ ($l+1$ rows in total). To fix $k$, we have to make sure
that the tableau has exactly $l(l+1)$ (twice the number of boxes in the
tableau of $V_{\rho}$). So, $k=l-1$ and we should represent
$V_{(l+1)\omega_{1}}$ by the tableau $\nu$ of shape $(2l,l-1,\dots,l-1)$
($l+1$ rows).

Finally, applying the Littlewood–Richardson rule, the multiplicity of
$V_{(l+1)\omega_{1}}$ is equal to the number of Littlewood–Richardson tableaux
of shape $\nu/\lambda$ and of weight $\lambda$. It is easy to see that the
first row of $\nu/\lambda$ should be filled by $1$. The next empty box appears
in the 3rd row, and it should be filled by $2$ (otherwise we would not be able
to fill in the 3rd column by the strictly increasing sequence). In the same
way, it is easy to see that the 4th row should be filled by $2$ and $3$ and so
on. So, we have a unique Littlewood–Richardson tableaux of shape $\nu/\lambda$
and of weight $\lambda$. See, for example, the picture below for $l=4$:

$\young(\bullet\bullet\bullet\bullet 1111,\bullet\bullet\bullet,\bullet\bullet
2,\bullet 23,234)$

∎

So, according to Proposition 11 the cohomology
$H^{1}_{+}(\mathfrak{g}_{-},\hat{\mathfrak{g}}^{\perp})$ does not vanish in
this case. This leads to the conjecture that the rational homogeneous variety
corresponding to the representation $V_{\rho}$ is non-rigid.

##  6\. Equivalence problem of extrinsic geometries - general case

###  6.1. Generalizations to the non-principal and to the infinite-dimensional
cases

In this section we extend the discussions in section 4 to obtain the
invariants of extrinsic geometries in more general settings.

First we drop the assumption that $L/L^{0}$ is embedded in a flag variety
$\operatorname{Flag}(V,\phi)$ since it is not the representation itself of $L$
on $V$ but the filtered structure on $L/L^{0}$ that matters.

Second we will not assume the conditions of $\overline{G}^{0}$\- invariance
(C1) and (C2) which were needed for “principal reduction” to obtain extrinsic
Cartan connections. Here we perform rather “step reduction” which does not
necessarily lead to Cartan connections, but still gives a general algorithm to
obtain the complete invariants of an extrinsic geometry.

Third we extend our discussions to the cases when $L$ may be infinite
dimensional. This generalization will have interesting applications to
extrinsic geometries of non-linear differential equations with respect to
infinite dimensional Lie groups such as the contact transformation groups.

To generalize to infinite-dimensional transformation groups, we rather treat
infinitesimal transformations.

######  Definition 9 ([31, 20]).

_A transitive Lie algebra sheaf $($of infinitesimal transformations$)$ on a
filtered manifold $(N,\mathfrak{f})$_ is a subsheaf $\mathcal{L}$ (of Lie
algebras) of the Lie algebra sheaf $\underline{TN}$ satisfying:

  1. i)

$\mathcal{L}$ leaves invariant $\mathfrak{f}$, that is,
$[\mathcal{L},\underline{\mathfrak{f}}^{p}]\subset\underline{\mathfrak{f}}^{p}$
for all $p\in\mathbb{Z}$;

  2. ii)

$\mathcal{L}$ is transitive, that is, the evaluation map
$\mathcal{L}_{x}\ni[X]_{x}\mapsto X_{x}\in T_{x}N$ is surjective for all $x\in
N$;

  3. iii)

$\mathcal{L}$ is defined by an involutive system of differential equations.

Note that the condition iii) means that $\mathcal{L}$ is “continuous” in the
sense of Lie and Cartan. For the notion of involutive differential equations
on filtered manifolds, see [22].

Let $\mathcal{P}(\mathcal{L})$ denote the pseudo-group of local
transformations of $(N,\mathfrak{f})$ generated by $\mathcal{L}$, namely by
all local flows $f_{t}$ of all sections $X$ of $\mathcal{L}$.

Let $\varphi_{1}\colon(M_{1},\mathfrak{f}_{M_{1}})\to(N,\mathfrak{f}_{N})$ and
$\varphi_{2}\colon(M_{2},\mathfrak{f}_{M_{2}})\to(N,\mathfrak{f}_{N})$ be two
morphisms of filtered manifolds, and let $x_{1}\in M_{1},$ $x_{2}\in M_{2}.$
We say that the germ of $\varphi_{1}$ at $x_{1}$ is $\mathcal{L}$-equivalent
to the germ of $\varphi_{2}$ at $x_{2}$, if there exist a local isomorphism
$h\colon(M_{1},\mathfrak{f}_{M_{1}})\to(M_{2},\mathfrak{f}_{M_{2}})$ with
$h(x_{1})=x_{2}$ and $a\in\mathcal{P}(\mathcal{L})$ such that
$a\circ\varphi_{1}=\varphi_{2}\circ h$.

It is under this $\mathcal{L}$-equivalence that we are going to study the
equivalence problem. Before going further, let us recall some basic facts
about our $\mathcal{L}$.

First of all we note that each stalk $\mathcal{L}_{x}$ has a natural
filtration $\\{\mathcal{L}^{p}_{x}\\}_{p\in\mathbb{Z}}$ induced from that of
$(N,\mathfrak{f}_{N})$ as defined by

$\mathcal{L}_{x}^{p}=\left\\{\begin{array}[]{ll}\\{[X]_{x}\in\mathcal{L}_{x}\mid
X_{x}\in\mathfrak{f}^{p}_{x}\\}&(p\leq 0)\\\\[2.84526pt]
\\{\xi\in\mathcal{L}^{p-1}_{x}\mid[\xi,\mathcal{L}^{i}_{x}]\subset\mathcal{L}^{i+p}_{x}\text{
for all }i<0\\}&(p\geq 1).\end{array}\right.$

Then it is easy to verify that

$[\mathcal{L}^{p}_{x},\mathcal{L}^{q}_{x}]\subset\mathcal{L}^{p+q}_{x}.$

Thus, $(\mathcal{L}_{x},\\{\mathcal{L}^{p}_{x}\\})$ is a filtered Lie algebra,
and passing to the quotient we get a graded Lie algebra
$\operatorname{gr}\mathcal{L}_{x}$. As shown later,
$\operatorname{gr}\mathfrak{f}_{x}$ is anti-isomorphic to
$\operatorname{gr}_{-}\mathcal{L}_{x}(=\sum_{p<0}\operatorname{gr}_{p}\mathcal{L}_{x})$.

With the notion of weighted jet bundle the filtration
$\\{\mathcal{L}^{p}_{x}\\}$ can be defined alternatively as follows. Regarding
$TN$ as a filtered vector bundle over a filtered manifold
$(N,\mathfrak{f}_{N})$, let $\widehat{J}^{(k)}TN$ be the weighted $k$-th jet
bundle. Denote by $\hat{j}^{(k)}_{x}$ the canonical map
$\underline{TN}_{\,x}\to\underline{\widehat{J}^{(k)}TN}_{\,x}$ for $x\in N$.
Then we have

$\mathcal{L}^{p+1}_{x}=\\{[X]_{x}\in\mathcal{L}_{x}\mid\hat{j}^{(p)}_{x}X=0\\}.$

Now let

$\displaystyle\mathcal{L}^{(k)}_{x}$
$\displaystyle=\\{\hat{j}^{(k)}_{x}X\mid[X]_{x}\in\mathcal{L}_{x}\\}=\mathcal{L}_{x}/\mathcal{L}_{x}^{k+1},$
$\displaystyle\mathcal{L}^{(k)}$ $\displaystyle=\bigcup_{x\in
N}\mathcal{L}^{(k)}_{x}.$

Then $\mathcal{L}^{(k)}\subset\widehat{J}^{(k)}TN$ may be regarded as a system
of differential equations defining $\mathcal{L}$.

The condition iii) of the definition of Lie algebra sheaf may be understood to
mean that there exists $k_{0}$ such that $\mathcal{L}^{(k_{0})}$ is weightedly
involutive and $\mathcal{L}^{(l)}$ is the prolongation of
$\mathcal{L}^{(k_{0})}$ for $l>k_{0}$. From the formal theory of differential
equations, it is equivalent to saying that $\mathcal{L}^{(k)}$ are vector
bundles for all $k$.

Taking the projective limit, we set

$\mathcal{L}^{(\infty)}_{x}=\lim_{\longleftarrow}\mathcal{L}^{(k)}_{x},$

which carries the structure of filtered Lie algebra and is called the formal
algebra of $\mathcal{L}$ at $x$. Note that from the transitivity of
$\mathcal{L}$ it follows that $\mathcal{L}^{(\infty)}_{x}$ and
$\mathcal{L}^{(\infty)}_{y}$ are isomorphic for all $x,y$ in a connected
component of $N$. Assuming $N$ connected if necessary, we fix a filtered Lie
algebra $\mathfrak{l}$, which is anti-isomorphic to all
$\mathcal{L}^{(\infty)}_{x}$, $x\in N$.

A proper generalization of the bundle $L\to L/L^{0}$ to infinite dimension may
be to introduce the following principal bundle equipped with a Pfaff system:

######  Theorem 3.

There is associated to a transitive Lie algebra sheaf $\mathcal{L}$ on a
filtered manifold $(N,\mathfrak{f})$ a principal fibre bundle $\mathfrak{L}$
over $N$ with structure group $L^{0}$ and $\mathfrak{l}$-valued 1-form
$\omega_{\mathfrak{L}}$ on $\mathfrak{L}$ such that

  1. i)

$L^{0}$ is a possibly infinite dimensional Lie group equipped with a
filtration $\\{L^{p}\\}_{p\geq 0}$ consisting of closed normal subgroups of
$L^{0}$ with the Lie algebra of $L^{p}$ being $\mathfrak{l}^{p}$ and

$\displaystyle L^{0}$
$\displaystyle=L^{(\infty)}=\lim_{\longleftarrow}L^{(k)},$
$\displaystyle\mathfrak{L}$
$\displaystyle=\mathfrak{L}^{(\infty)}=\lim_{\longleftarrow}\mathfrak{L}^{(k)},$

where $L^{(k)}=L^{0}/L^{k+1}$, $\mathfrak{L}^{(k)}=\mathfrak{L}/L^{k+1}$.

  2. ii)

The map $\big{(}\omega_{\mathfrak{L}}\big{)}_{z}\colon
T_{z}\mathfrak{L}\to\mathfrak{l}$ is a filtration preserving isomorphism for
all $z\in\mathfrak{L}$.

  3. iii)

$\langle\omega_{\mathfrak{L}},\tilde{A}\rangle=A$ for all
$A\in\mathfrak{l}^{0}$.

  4. iv)

The adjoint action of $L^{0}$ on $\mathfrak{l}^{0}$ is extended to
$\mathfrak{l}$ and

$R^{*}_{a}\omega_{\mathfrak{L}}=\operatorname{Ad}(a)^{-1}\omega_{\mathfrak{L}}\quad\text{for
all }a\in L^{0}.$

  5. v)

$d\omega_{\mathfrak{L}}+1/2[\omega_{\mathfrak{L}},\omega_{\mathfrak{L}}]=0$.

  6. vi)

If $X$ is a section of $\mathcal{L},$ then there is a unique lift
$\widehat{X}$ of $X$ to $\mathfrak{L}$ such that
$L_{\widehat{X}}\omega_{\mathfrak{L}}=0$. Conversely, if a local vector field
$Z$ on $\mathfrak{L}$ satisfies $L_{Z}\omega_{\mathfrak{L}}=0$, then there
exists a section $X$ of $\mathcal{L}$ such that $Z=\widehat{X}$.

Note that $\mathfrak{L}$ and $L^{0}$ can be infinite dimensional. But since
they are projective limits of finite dimensional objects, we can treat those
infinite dimensional objects similarly to the finite dimensional case.

###### Proof.

To prove the theorem we employ the scheme developed in [21].

First we note that since $\mathcal{L}$ is transitive on $N$,
$(N,\mathfrak{f}_{N})$ has a constant symbol, say $\mathfrak{m}$. Let
$\mathcal{R}^{(k)}(N,\mathfrak{f}_{N},\mathfrak{m})$ or simply
$\mathcal{R}^{(k)}$ be the reduced frame bundle of order $(k+1)$ and set
$\mathcal{R}=\mathcal{R}^{(\infty)}={\displaystyle\lim_{\longleftarrow}\mathcal{R}^{(k)}}$
(see [21, p.316]). Recall that every automorphism $h$ of
$(N,\mathfrak{f}_{N})$ is uniquely lifted to an automorphism $\hat{h}^{(k)}$
of $\mathcal{R}^{(k)}$. Therefore we have a lift
$\widehat{\mathcal{P}(\mathcal{L})}^{(k)}$ of $\mathcal{P}(\mathcal{L})$ to
$\mathcal{R}^{(k)}$. Passing to the infinitesimal, every infinitesimal
automorphism $X$ of $(N,\mathfrak{f}_{N})$ is uniquely lifted to the one
$\widehat{X}^{(k)}$ on $\mathcal{R}^{(k)}$ and the Lie algebra sheaf
$\mathcal{L}$ to the one $\widehat{\mathcal{L}}^{(k)}$ on $\mathcal{R}^{(k)}$.
Clearly, we have
$\mathcal{P}(\widehat{\mathcal{L}}^{(k)})=\widehat{\mathcal{P}(\mathcal{L})}^{(k)}$.

By evaluation, $\widehat{\mathcal{L}}^{(k)}$ defines a subbundle
$D^{(k)}\subset T\mathcal{R}^{(k)}$, which is, as easily seen, of constant
rank and completely integrable. Now choose
$\mathring{z}^{(k)}\in\mathcal{R}^{(k)}$ ($k=0,1,\dots,$) such that
$\pi^{k+1}_{k}(\mathring{z}^{(k+1)})=\mathring{z}^{(k)}$, where
$\pi^{k+1}_{k}$ denotes the projection
$\mathcal{R}^{(k+1)}\to\mathcal{R}^{(k)}$. Let $\mathfrak{L}^{(k)}$ be the
maximal integral manifold of $D^{(k)}$ through $\mathring{z}^{(k)}$, and set

$\mathfrak{L}=\lim_{\longleftarrow}\mathfrak{L}^{(k)},\quad\mathring{z}=\lim_{\longleftarrow}\mathring{z}^{(k)}.$

Then we see immediately that $\mathfrak{L}^{(k+1)}\to\mathfrak{L}^{(k)}$ is a
surjective submersion for all $\infty\geq k\geq-1$, where we set
$\mathfrak{L}^{-1}=N$. Not only this, we are now going to see that
$\mathfrak{L}^{(k)}\to\mathfrak{L}^{(j)}$ are all principal fibre bundles for
$k\geq j$.

Let $\omega_{\mathcal{R}}$ be the canonical 1-form on $\mathcal{R}$, which is
a 1-form on $\mathcal{R}$ taking values in
$E(\mathfrak{m})=\mathfrak{m}\oplus\mathfrak{g}^{0}(\mathfrak{m})$, where
$G^{0}(\mathfrak{m})$ is the structure group of $\mathcal{R}\to N$ and
$\mathfrak{g}^{0}(\mathfrak{m})$ is the Lie algebra of $G^{0}(\mathfrak{m})$.

Let $\omega_{\mathfrak{L}}$ be the restriction of $\omega_{\mathcal{R}}$ to
$\mathfrak{L}$: $\omega_{\mathcal{L}}=\iota^{*}\omega_{\mathcal{R}}$, where
$\iota$ denotes the inclusion $\mathfrak{L}\to\mathcal{R}$. If
$\Phi\in\widehat{\mathcal{P}(\mathcal{L})}$, then
$\Phi^{*}\omega_{\mathcal{R}}=\omega_{\mathcal{R}}$ and
$\Phi(\mathfrak{L})=\mathfrak{L}$. Therefore
$\Phi^{*}\omega_{\mathfrak{L}}=\omega_{\mathfrak{L}}$. It then follows that
there is a subspace $E_{\mathfrak{L}}\subset E(\mathfrak{m})$ such that we
have an isomorphism

$\big{(}\omega_{\mathfrak{L}})_{z}\colon T_{z}\mathfrak{L}\to
E_{\mathfrak{L}}\quad\text{for all }z\in\mathfrak{L},$

and we have

$d\omega_{\mathfrak{L}}+1/2\gamma_{\mathfrak{L}}(\omega_{\mathfrak{L}},\omega_{\mathfrak{L}})=0$

with a constant structure function
$\gamma_{\mathfrak{L}}\in\operatorname{Hom}(\wedge^{2}E_{\mathfrak{L}},E_{\mathfrak{L}})$,
because $\widehat{\mathcal{P}(\mathcal{L})}$ is transitive on $\mathfrak{L}$.

We see therefore that $\gamma_{\mathfrak{L}}\colon E_{\mathfrak{L}}\times
E_{\mathfrak{L}}\to E_{\mathfrak{L}}$ defines a Lie algebra structure on
$E_{\mathfrak{L}}$. Moreover, if we set

$\mathfrak{g}^{0}(\mathfrak{L})=\mathfrak{g}^{0}(\mathfrak{m})\cap
E_{\mathfrak{L}},$

the Lie algebra structure defined by
$\gamma_{\mathfrak{L}}|_{\mathfrak{g}^{0}(\mathfrak{L})\times\mathfrak{g}^{0}(\mathfrak{L})}$
coincides with the one induced from $\mathfrak{g}^{0}(\mathfrak{m})$.

Let $X,Y\in\mathcal{L}$. Substituting $\hat{X}$, $\hat{Y}$ into the structure
equation, we have

$\hat{X}\omega_{\mathfrak{L}}(\hat{Y})-\hat{Y}\omega_{\mathfrak{L}}(\hat{X})-\omega_{\mathfrak{L}}([\hat{X},\hat{Y}])+\gamma_{\mathfrak{L}}\big{(}\omega_{\mathfrak{L}}(\hat{X}),\omega_{\mathfrak{L}}(\hat{Y})\big{)}=0.$

But since $L_{\hat{X}}\omega_{\mathfrak{L}}=0$, we have

$\hat{X}\omega(\hat{Y})=(L_{\hat{X}}\omega_{\mathfrak{L}})(\hat{Y})+\omega_{\mathfrak{L}}(L_{\hat{X}}\hat{Y})=\omega_{\mathfrak{L}}([\hat{X},\hat{Y}]).$

Therefore we have

$\omega_{\mathfrak{L}}([\hat{X},\hat{Y}])+\gamma_{\mathfrak{L}}\Big{(}\omega_{\mathfrak{L}}(\hat{X}),\omega_{\mathfrak{L}}(\hat{Y})\Big{)}=0,$

which implies that the map

$\mathfrak{L}_{x}\xrightarrow{\widehat{\phantom{X}}}\widehat{\mathcal{L}}_{z}\xrightarrow{\left(\omega_{\mathfrak{L}}\right)_{z}}E_{\mathfrak{L}}\quad(z\in\mathfrak{L},x=\pi_{N}(z)\in
N)$

gives an anti-isomorphism
$\mathcal{L}^{(\infty)}_{x}\xrightarrow{\omega^{\wedge}_{x}}(E_{\mathfrak{L}},\gamma_{\mathfrak{L}})$
of Lie algebras, where $\mathcal{L}_{x}^{(\infty)}$ denotes the formal algebra
of $\mathcal{L}$ at $x$.

Recalling that $E_{\mathfrak{L}}$ and $\mathcal{L}_{x}^{(\infty)}$ have
natural filtrations $\\{\phi^{p}E_{\mathfrak{L}}\\}$ and
$\\{\phi^{p}\mathcal{L}_{x}^{(\infty)}\\}$, we remark that
$\omega^{\wedge}_{x}\Big{(}\phi^{p}\mathcal{L}_{x}^{(\infty)}\Big{)}=\phi^{p}E_{\mathfrak{L}}$
for all $p\in\mathbb{Z}$. In fact, it is easy to see that it holds for $p\leq
0$. But in both filtrations the space $\phi^{p}$ ($p\geq 0$) is determined by

$\phi^{p}=\\{X\in\phi^{p-1}\mid[X,\phi^{a}]\subset\phi^{p+a}\text{ for all
}a<0\\}.$

Therefore the assertion is valid for all $p\in\mathbb{Z}$.

In particular, we have

  1. (1)

$\operatorname{gr}\mathfrak{f}_{N,x}$ is anti-isomorphic to
$\operatorname{gr}_{-}\mathcal{L}_{x}$ for $x\in N$.

  2. (2)

For $X\in\mathcal{L}_{x}$, $z\in\mathfrak{L}$ with $\pi_{N}(z)=x$, and
$k\in\mathbb{Z}$, $j^{k}_{x}X=0$ if and only if $\hat{X}_{z}\equiv
0\mod\phi^{k+1}T_{z}\mathfrak{L}$.

In relation to (2), in general, we can prove that if $X$ is an infinitesimal
automorphism of a filtered manifold $(M,\mathfrak{f})$, then, for $k<0$,
$j^{k}_{x}X=0$ if and only if $X_{k}\in\mathfrak{f}^{k+1}_{x}$.

Let us show that if $za,zb\in\mathfrak{L}$ for $z\in\mathfrak{L}$, $a,b\in
G^{0}(\mathfrak{m})$, then $zab\in\mathfrak{L}$. In fact, since
$z,za\in\mathfrak{L}$ and $\widehat{\mathcal{P}}(\mathcal{L})$ is transitive
on $\mathfrak{L}$, there exist neighborhoods $U,U^{\prime}$ of
$x=\pi^{\mathcal{R}}(z)$ in $N$ and a diffeomorphism
$\Phi\colon(\pi^{\mathcal{R}})^{-1}(U)\to(\pi^{\mathcal{R}})^{-1}(U^{\prime})$
such that $\Phi^{*}\omega_{\mathcal{R}}=\omega_{\mathcal{R}}$,
$\Phi(wg)=\Phi(w)g$, $w\in(\pi^{\mathcal{R}})^{-1}(U)$, $g\in
G^{0}(\mathfrak{m})$,
$\Phi((\pi^{\mathcal{R}})^{-1}(U)\cap\mathfrak{L})\subset(\pi^{\mathcal{R}})^{-1}(U^{\prime})\cap\mathfrak{L}$
and $\Phi(z)=za$, where $\pi^{\mathcal{R}}$ denotes the projection
$\mathcal{R}\to N$. Then since
$zb\in\mathfrak{L}\cap(\pi^{\mathcal{R}})^{-1}(U)$, we have
$zab=\Phi(z)b=\Phi(zb)\in\mathfrak{L}$, which proves the assertion.

Now it is easy to see that $\mathfrak{L}$ is a principal fibre bundle over $N$
with structure group $G^{0}(\mathfrak{L})$, where $G^{0}(\mathfrak{L})$ is a
Lie subgroup of $G^{0}(\mathfrak{m})$ with Lie algebra
$\mathfrak{g}^{0}(\mathfrak{L})$, which can be identified with
$\mathfrak{l}^{0}$.

It is not difficult to verify that all the statements in Theorem 3 hold. ∎

######  Remark 3.

The statement vi) of Theorem 3 means that
$(\mathfrak{L},\omega_{\mathfrak{L}})$ can be regarded as a defining equation
of $\mathcal{L}$, in other words, an infinitesimal automorphism $X$ of
$(N,\mathfrak{f})$ is a section of $\mathcal{L}$ if and only if the lift
$\hat{X}$ of $X$ to $\mathcal{R}$ is tangent to $\mathfrak{L}$, which is, in
turn, equivalent to saying that the lift $\hat{X}^{(k_{0})}$ of $X$ to
$\mathcal{R}^{(k_{0})}$ is tangent to $\mathfrak{L}^{(k_{0})}$ for certain
integer $k_{0}$.

This integer $k_{0}$ is given by the condition:

$H^{1}_{r}(\operatorname{gr}_{-}\mathfrak{l},\operatorname{gr}\mathfrak{l})=0\quad\text{
for }r\geq k_{0}.$

The existence of such $k_{0}$ is assured by the finitude of this cohomology
group (see [20]).

It being prepared, let
$\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$ be a morphism. Then
via pull-back we obtain the principal fibre bundle $\varphi^{*}\mathfrak{L}$
over $M$ with structure group $L^{0}$ endowed with an $\mathfrak{l}$-valued
1-form $\tilde{\varphi}^{*}\omega$, where
$\tilde{\varphi}\colon\varphi^{*}\mathfrak{L}\to\mathfrak{L}$ is a canonical
inclusion. The bundle with Pfaff forms
$(\varphi^{*}\mathcal{L},\tilde{\varphi}^{*}\omega)$, denoted also by
$(Q_{\varphi},\omega_{\varphi})$, is called an extrinsic bundle associated
with $\varphi$.

¿From now on we assume
$\operatorname{gr}\varphi_{*,x}\colon\operatorname{gr}(\mathfrak{f}_{M})_{x}\to\operatorname{gr}(\mathfrak{f}_{N})_{\varphi(x)}$
is injective for all $x\in M$, so that $\operatorname{gr}\varphi_{*,x}$ is an
injective graded Lie algebra homomorphism.

Let $\mathfrak{g}_{-}=\bigoplus_{p<0}\mathfrak{g}_{p}$ be a graded subalgebra
of $\operatorname{gr}\mathfrak{l}_{-}=\bigoplus_{p<0}\mathfrak{l}$.

######  Definition 10.

We say that $\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$ has
constant symbol of type $(\mathfrak{g}_{-},\mathcal{L}),$ if for any $x,y\in
M$ there exist $a\in\mathcal{P}(\mathcal{L})$ and isomorphisms of graded Lie
algebras $\beta_{x}$ and $\beta_{y}$ which make the following diagram
commutative:

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Then we have

######  Proposition 13.

The map $\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$ is of type
$(\mathfrak{g}_{-},\mathcal{L}),$ if and only if
$(Q_{\varphi},\omega_{\varphi})$ satisfies the following criterion:

For any $x\in M$ there exists $z\in Q_{\varphi}$ over $x$ such that
$\operatorname{gr}\omega_{z}(\operatorname{gr}(\mathfrak{f}_{M})_{x})=\mathfrak{g}_{-}$.

Now we are going to construct the invariants of extrinsic geometries
$\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$ under
$\mathcal{L}$. To avoid surplus complexity, we assume:

  1. (C0)

$\mathfrak{l}=\widehat{\bigoplus}\,\mathfrak{l}_{p}$, that is, $\mathfrak{l}$
is isomorphic to the completion of $\operatorname{gr}\mathfrak{l}$.

We also restrict ourselves to morphisms $\varphi$ of constant symbol, say of
type $(\mathfrak{g}_{-},\mathcal{L})$.

We follow the discussions in section 4 by making necessary modifications to
adapt to this general case.

Let $\bar{\mathfrak{g}}=\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})$ be
the relative prolongation of $\mathfrak{g}_{-}$ in $\mathfrak{l}$. Similarly
as before we define

$\overline{G}_{0}=\\{a\in
L^{(0)}\mid\operatorname{Ad}(a)\mathfrak{g}_{-}=\mathfrak{g}_{-}\\}$

and $\overline{G}(k)$, $\bar{\mathfrak{g}}(k)$ as well as their filtrations
$\\{\overline{G}(k)^{p}\\}$, $\\{\bar{\mathfrak{g}}(k)^{p}\\}$. Recall thus
for instance,

$\displaystyle\bar{\mathfrak{g}}(k)$
$\displaystyle=\bar{\mathfrak{g}}(k)^{0}=\bigoplus_{0\leq p\leq
k}\bar{\mathfrak{g}}_{p}\oplus\bigoplus_{q>k}\mathfrak{l}_{q},$
$\displaystyle\ \bar{\mathfrak{g}}(k)^{k}$
$\displaystyle=\bar{\mathfrak{g}}_{k}\oplus\bigoplus_{q>k}\mathfrak{l}_{q}.$

We chose complementary subspaces
$\bar{\mathfrak{g}}^{\prime}=\bigoplus\mathfrak{g}^{\prime}_{p}$ to
$\bar{\mathfrak{g}}$ in $\mathfrak{l}$ so that

$\mathfrak{l}_{p}=\mathfrak{g}_{p}\oplus\mathfrak{g}^{\prime}_{p},$

and we have a direct sum decomposition:

$\mathfrak{l}=E(k)\oplus E^{\prime}(k)$

with

$E(k)=\mathfrak{g}_{-}\oplus\bar{\mathfrak{g}}(k),\
E^{\prime}(k)=\bigoplus_{q\leq k}\mathfrak{g}^{\prime}_{q}.$

We choose also complementary subspace $W=\bigoplus_{p\geq 1}W_{p}$ such that
we have for all $p\geq 1$

$\operatorname{Hom}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})_{p}=W_{p}\oplus\partial(\mathfrak{l}/\bar{\mathfrak{g}})_{p}.$

Let $(Q_{\varphi},\omega_{\varphi})$ be the original extrinsic bundle induced
from $\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$.

First we define $(Q(0),\omega(0))$ by

$Q(0)=\\{z\in
Q_{\varphi}\mid\operatorname{gr}(\omega_{\varphi})_{z}\big{(}\operatorname{gr}(\mathfrak{f}_{M})_{x}\big{)}=\mathfrak{g}_{-}\\}$

and $\omega(0)=\iota^{*}\omega_{\varphi}$, where $\iota\colon Q(0)\to
Q_{\varphi}$ is the canonical inclusion.

Now we construct inductively

$Q(k)\to B(k-1),\quad\omega(k),\quad\chi(k),$

in such a way that:

  1. i)

$Q(k)$ is a principal fibre bundle over the base space

$B(k-1)=Q(k-1)^{(k-1)}=Q(k-1)/\bar{G}(k)^{k}$

with structure group $\overline{G}(k)^{k}$. Furthermore there is a canonical
inclusion $\iota\colon Q(k)\to Q(k-1)$ as principal fibre bundles over
$B(k-1)$,

  2. ii)

$\omega(k)=\iota^{*}\omega(k-1)$ and

$R_{a}^{*}\omega(k)=\operatorname{Ad}(a)^{-1}\omega(k)\quad\text{for}\
a\in\bar{G}(k)^{k},$

  3. iii)

If we write $\omega(k)_{II}=\chi(k)\omega_{I}$ according to the direct sum
decomposition $\mathfrak{l}=E(k)\oplus E^{\prime}(k)$, then

$\omega(k)_{I}\colon T_{z}Q(k)\to E(k)$

is a filtration preserving linear isomorphism for all $z\in Q(k)$,

  4. iv)

Define a map $\chi(k)\colon Q(k)\to\operatorname{Hom}(E(k),E^{\prime}(k))$ by
$\omega(k)_{II}=\chi(k)\omega(k)_{I}$ and decompose it as

$\chi(k)=\chi^{-}(k)+\chi^{+}(k)\quad\text{and}\
\chi(k)_{p}=\chi^{-}(k)_{p}+\chi^{+}(k)_{p}$

by requiring that

$\displaystyle\chi^{-}(k)_{p}\colon Q(k)\to$
$\displaystyle\operatorname{Hom}(\mathfrak{g}_{-},E^{\prime}(k))_{p},$
$\displaystyle\chi^{+}(k)_{p}\colon Q(k)\to$
$\displaystyle\operatorname{Hom}(\bigoplus_{0\leq i\leq
k}\bar{\mathfrak{g}}_{i},E^{\prime}(k))_{p},$ $\displaystyle\chi(k)=$
$\displaystyle\sum\chi(k)_{p},$ $\displaystyle\chi(k)^{-}=$
$\displaystyle\sum\chi(k)^{-}_{p},$ $\displaystyle\chi(k)^{+}=$
$\displaystyle\sum\chi(k)^{+}_{p}.$

Then

    1. a)

$\chi(k)_{i}=\iota^{*}\chi(k-1)_{i}$ for $i<k$,

    2. b)

$\chi(k)_{p}=0$ for $p\leq 0$,

    3. c)

$\chi^{-}(k)_{k}\in W_{k}$.

${\mathcal{L}}$${Q_{\varphi}}$${Q(0)}$${Q(1)}$${Q(2)}$${\cdots}$${Q(k)}$${Q(k+1)}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${Q(k)^{(k+1)}}$${Q(k+1)^{(k+1)}}$${B(k+1)}$${Q(k)^{(k)}}$${B(k)}$${\vdots}$${\vdots}$${B(k-1)}$${\vdots}$${Q(1)^{(2)}}$${Q(2)^{(2)}}$${B(2)}$${\vdots}$${\vdots}$${Q(0)^{(1)}}$${Q(1)^{(1)}}$${B(1)}$${L^{(0)}}$${Q_{\varphi}^{(0)}}$${Q(0)^{(0)}}$${B(0)}$${N}$${M}$${M}$$\scriptstyle{\overline{G}_{k+1}}$$\scriptstyle{\overline{G}_{k}}$$\scriptstyle{\overline{G}_{2}}$$\scriptstyle{\overline{G}_{1}}$$\scriptstyle{\overline{G}_{0}}$$\scriptstyle{\varphi}$

Note the main differences from the previous case are first of all that $Q(k)$
is no more a principal bundle over $M$ but only over $Q(k-1)^{(k-1)}$ and
second that $\chi^{+}(k)$ here does not vanish, in general.

The construction is based on the following formula:

######  Proposition 14.

$R_{\exp A_{k+1}}^{*}\chi^{-}(k)_{k+1}=\chi^{-}(k)_{k+1}-\partial
A_{k+1}\quad\text{for}\
A_{k+1}\in\bar{\mathfrak{g}}(k)_{k+1}(=\mathfrak{l}_{k+1}).$

###### Proof.

We have already shown this formula in section 4 under the assumptions of
$\bar{G}^{0}$ invariance conditions (C1), (C2), but to prove it in our general
case we should argue more carefully.

In the formula $\partial$ denotes the coboundary operator in the complex:

$0\rightarrow\mathfrak{l}/\bar{\mathfrak{g}}\overset{\partial}{\rightarrow}\operatorname{Hom}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})\rightarrow\dots,$

identifying $\bar{\mathfrak{g}}^{\prime}$ with
$\mathfrak{l}/\bar{\mathfrak{g}}$, and writing simply $\partial$ for
$\partial\circ\pi$, where $\pi$ is the projection:
$\mathfrak{l}\to\mathfrak{l}/\bar{\mathfrak{g}}$. During the proof, we simply
write

$\omega(k)=\omega,\quad\chi(k)=\chi=\sum_{i\geq
1}(\chi^{+}_{i}+\chi^{-}_{i}),\quad\alpha=\exp A_{k+1}$

Recall that

$\displaystyle R_{\alpha}^{*}\omega$
$\displaystyle=\operatorname{Ad}(\alpha^{-1})\omega,$ $\displaystyle\omega$
$\displaystyle=\omega_{I}+\omega_{II},$ $\displaystyle\omega_{II}$
$\displaystyle=\chi\omega_{I}.$

Then we have

$\displaystyle R_{\alpha}^{*}(\omega_{I}+\omega_{II})=$
$\displaystyle\operatorname{Ad}(\alpha^{-1})(\omega_{I}+\omega_{II})$
$\displaystyle=$
$\displaystyle\operatorname{Ad}(\alpha^{-1})^{I}_{I}\omega_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\omega_{II}+\operatorname{Ad}(\alpha^{-1})^{I}_{II}\omega_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\omega_{II}$
$\displaystyle=$
$\displaystyle\operatorname{Ad}(\alpha^{-1})^{I}_{I}\omega_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\chi\omega_{I}+\operatorname{Ad}(\alpha^{-1})^{I}_{II}\omega_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi\omega_{I}$
$\displaystyle=$
$\displaystyle(\operatorname{Ad}(\alpha^{-1})^{I}_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\chi)\omega_{I}+(\operatorname{Ad}(\alpha^{-1})^{I}_{II}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi)\omega_{I},$

where we use the notation $\operatorname{Ad}(\alpha^{-1})^{I}_{II}$ to denote
the $\operatorname{Hom}(E(k),E^{\prime}(k))$-component of
$\operatorname{Ad}(\alpha^{-1})$ and similar notations.

Therefore we have

$\displaystyle R_{\alpha}^{*}\omega_{I}=$
$\displaystyle\Big{(}\operatorname{Ad}(\alpha^{-1})^{I}_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\chi\Big{)}\omega_{I},$
$\displaystyle R_{\alpha}^{*}\omega_{II}=$
$\displaystyle\Big{(}\operatorname{Ad}(\alpha^{-1})^{I}_{II}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi\Big{)}\omega_{I}.$

On the other hand, we have

$R_{\alpha}^{*}\omega_{II}=(R_{\alpha}^{*}\chi)(R_{\alpha}^{*}\omega_{I}).$

Therefore we have

$\Big{(}\operatorname{Ad}(\alpha^{-1})^{I}_{II}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi\Big{)}\omega_{I}=(R_{\alpha}^{*}\chi)\Big{(}\operatorname{Ad}(\alpha^{-1})^{I}_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\chi\Big{)}\omega_{I}.$

Letting $\omega_{I}$ take values in $\mathfrak{g}_{-}$, we have

$\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{II}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi^{-}=(R_{\alpha}^{*}\chi)\Big{(}\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I}+\operatorname{Ad}(\alpha^{-1})^{II}_{I}\chi^{-}\Big{)},$

and then

$\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{II}+\operatorname{Ad}(\alpha^{-1})^{II}_{II}\chi^{-}=(R_{\alpha}^{*}\chi^{-})\Big{(}\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I^{-}}+\operatorname{Ad}(\alpha^{-1})^{II}_{I^{-}}\chi^{-}\Big{)}\\\
+(R_{\alpha}^{*}\chi^{+})\Big{(}\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I^{+}}+\operatorname{Ad}(\alpha^{-1})^{II}_{I^{+}}\chi^{-}\Big{)},$

where $\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I^{-}}$ denotes the
$\operatorname{Hom}(\mathfrak{g}_{-},\mathfrak{g}_{-})$-component of
$\operatorname{Ad}(\alpha^{-1})$ and the same convention for the similar
notation.

Now enter $v_{p}\in\mathfrak{g}_{p}\ (p<0)$ into the above formula and catch
the $(\mathfrak{l}/\bar{\mathfrak{g}})_{i+p}$ -component for $i\leq k+1$.
Taking into account that

$\displaystyle\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{II}=$
$\displaystyle-\operatorname{ad}(A_{k+1})^{I^{-}}_{II}+\text{higher order
terms}$ $\displaystyle\operatorname{Ad}(\alpha^{-1})^{II}_{II}=$
$\displaystyle\operatorname{id}_{II}-\operatorname{ad}(A_{k+1})^{II}_{II}+\text{higher
order terms}$

and that

$\Big{(}\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I^{+}}+\operatorname{Ad}(\alpha^{-1})^{II}_{I^{+}}\chi^{-}\Big{)}(v_{p})\in\mathfrak{l}_{k+1+p}+\mathfrak{l}_{k+2+p}+\cdots$

and therefore

$\Big{(}R_{\alpha}^{*}\chi^{+}_{j+p})(\operatorname{Ad}(\alpha^{-1})^{I^{-}}_{I^{+}}+\operatorname{Ad}(\alpha^{-1})^{II}_{I^{+}}\chi^{-}\Big{)}(v_{p})=0\quad\text{for
}j\leq k+1,$

we have

$\displaystyle R_{\alpha}^{*}\chi^{-}_{i}$
$\displaystyle=\chi^{-}_{i}\quad\text{for }i\leq k$ $\displaystyle
R_{\alpha}^{*}\chi^{-}_{k+1}$ $\displaystyle=\chi^{-}_{k+1}-\partial A_{k+1},$

which proves the proposition. ∎

We can then easily carry out the inductive construction of $Q(k),$ and by
passing to limit we obtain

$Q=Q(\infty),\quad\omega=\omega(\infty),\quad\chi=\chi(\infty).$

The structure function $\chi\colon
Q\to\operatorname{Hom}(\bar{\mathfrak{g}},\mathfrak{l}/\bar{\mathfrak{g}})$ is
an invariant of $Q$ and that of the initial morphism
$\varphi\colon(M,\mathfrak{f})\to(N,\mathfrak{f}_{N})$. To examine it more
closely, we write

$\displaystyle\chi$ $\displaystyle=\sum(\chi^{-}_{k}+\chi^{+}_{k}),$
$\displaystyle\chi^{+}_{k}$ $\displaystyle=\sum_{0\leq
i<k}\chi^{+i}_{k},\quad\text{where }\chi^{+i}_{k}\text{ takes its value in
}\operatorname{Hom}(\bar{\mathfrak{g}}_{i},\bar{\mathfrak{g}}_{k}).$

Then we have

######  Proposition 15.

$\displaystyle\partial\chi^{-}_{k+1}$
$\displaystyle=\Psi^{-}_{k+1}(\chi_{\ell},\ D\chi_{\ell}\ ;\ \ell\leq k),$
$\displaystyle\chi^{+}_{k+1}$ $\displaystyle=\Psi^{+}_{k+1}(\chi^{-}_{\ell},\
\chi^{+}_{m},\ D\chi^{-}_{\ell},\ D\chi^{+}_{m}\ ;\ \ell\leq k+1,\,m\leq k),$

where $\Psi^{-}_{k+1}$ is a vector valued function determined explicitly only
by $\chi_{\ell}$ $(\ell\leq k)$ and their derivatives, and $\Psi^{+}_{k+1}$ is
determined only by $\chi^{-}_{\ell}$ $(\ell\leq k+1),$ $\chi^{+}_{m}$ $(m\leq
k)$ and their derivatives.

Moreover $\Psi^{-}_{k+1}=0$ if $\chi_{\ell}=0$ $(\ell\leq k),$ and
$\Psi^{+}_{k+1}=0$ if $\chi_{\ell}=0$ $(\ell\leq k)$ and $\chi^{-}_{k+1}=0$.

###### Proof.

From the structure equation:

$\displaystyle d\omega+\frac{1}{2}[\omega,\omega]=0,$
$\displaystyle\omega=\omega_{I}+\omega_{II},$
$\displaystyle\omega_{II}=\chi\omega_{I},$

it follows that

$\displaystyle
d\omega_{I}+\frac{1}{2}[\omega_{I},\omega_{I}]+[\omega_{I},\omega_{II}]_{I}+\frac{1}{2}[\omega_{II},\omega_{II}]_{I}$
$\displaystyle=0,$ $\displaystyle
d\omega_{II}+[\omega_{I},\omega_{II}]_{II}+\frac{1}{2}[\omega_{II},\omega_{II}]_{II}$
$\displaystyle=0,$

and then

$\displaystyle
d\omega_{I}+\frac{1}{2}[\omega_{I},\omega_{I}]+[\omega_{I},\chi\omega_{I}]_{I}+\frac{1}{2}[\chi\omega_{I},\chi\omega_{I}]_{I}$
$\displaystyle=0,$ $\displaystyle
d(\chi\omega_{I})+[\omega_{I},\chi\omega_{I}]_{II}+\frac{1}{2}[\chi\omega_{I},\chi\omega_{I}]_{II}$
$\displaystyle=0,$

which yields

$[\omega_{I},\chi\omega_{I}]_{II}-\chi(\frac{1}{2}[\omega_{I},\omega_{I}])-\chi([\omega_{I},\chi\omega_{I}]_{I})\\\
+\frac{1}{2}[\chi\omega_{I},\chi\omega_{I}]_{II}-\chi(\frac{1}{2}[\chi\omega_{I},\chi\omega_{I}]_{I})+d\chi\wedge\omega_{I}=0.$

Now evaluating the above equation at $\tilde{v}_{p}\wedge\tilde{w}_{q}$, where
$\tilde{v},\tilde{w}$ denote the tangent vectors such that
$\omega_{I}(\tilde{v}_{p})=v_{p}\in\mathfrak{g}_{p}$,
$\omega_{I}(\tilde{w}_{q})=w_{q}\in\mathfrak{g}_{q}$ and $p,q<0$, and looking
at the $(\mathfrak{l}/\bar{\mathfrak{g}})_{k+1+p+q}$-component, we have

$(\partial\chi^{-}_{k+1})(v_{p},w_{q})=\mathcal{A}\sum\limits_{\begin{subarray}{c}a+b=k+1,\\\
a,b>0\end{subarray}}\Big{(}\chi^{-}_{a}[v_{p},\chi^{-}_{b}(w_{q})]_{I}+[\chi^{-}_{a}(v_{p}),\chi^{-}_{b}(w_{q})]_{II}\Big{)}\\\
+\mathcal{A}\sum\limits_{\begin{subarray}{c}a+b+c=k+1,\\\
a,b,c>0\end{subarray}}\chi^{-}_{a}[\chi^{-}_{b}(v_{p}),\chi^{-}_{c}(w_{q})]_{I}-\tilde{v}_{p}\chi^{-}_{k+1+p}(w_{q})+\tilde{w}_{q}\chi^{-}_{k+1+q}(v_{p})\\\
+\mathcal{A}\sum_{0<b<k+1}\chi^{+}_{k+1+p+q}[v_{p},\chi^{-}_{b}(w_{q})]_{I}+\mathcal{A}\sum\limits_{\begin{subarray}{c}b+c<k+1,\\\
b,c>0\end{subarray}}\chi^{+}_{k+1+p+q}[\chi^{-}_{b}(v_{p}),\chi^{-}_{c}(w_{q})]_{I},$

where $\mathcal{A}$ denotes the alternating sum in $v_{p}$, $w_{q}$. The last
formula defines explicitly the function $\Psi^{-}_{k+1}$.

Next we evaluate the previous formula at $\tilde{A}_{i}\wedge\tilde{v}_{p}$,
where $A_{i}\in\bar{\mathfrak{g}}_{i}$ ($i\geq 0$), $v_{p}\in\mathfrak{g}_{p}$
($p<0$) and $\tilde{A}_{i}$, $\tilde{v}_{p}$ are the corresponding tangent
vectors. Looking at $(\mathfrak{l}/\bar{\mathfrak{g}})_{k+1+p}$-component, we
have

$[v_{p},\chi^{+}_{k+1}A_{i}]_{II}=[A_{i},\chi^{-}_{k+1-i}v_{p}]_{II}-\chi^{-}_{k+1-i}[A_{i},v_{p}]-\chi^{+}_{k+1+p}[A_{i},v_{p}]\\\
-\sum_{a+b+i=k+1}\chi^{-}_{a}[A_{i},\chi^{-}_{b}v_{p}]_{I}-\sum_{b+i<k+1}\chi^{+}_{k+1+p}[A_{i},\chi^{-}_{b}v_{p}]_{I}\\\
+\sum_{a+b=k+1}\chi^{-}_{a}[v_{p},\chi^{+}_{b}A_{i}]_{I}+\sum_{b<k+1}\chi^{+}_{k+1+p}[v_{p},\chi^{+}_{b}A_{i}]_{I}\\\
+\sum_{a+b=k+1}[\chi^{+}_{a}A_{i},\chi^{-}_{b}v_{p}]_{II}-\sum_{a+b+c=k+1}\chi^{-}_{a}[\chi^{+}_{b}A_{i},\chi^{-}_{c}v_{p}]_{I}\\\
-\sum_{b+c<k+1}[\chi^{+}_{b}A_{i},\chi^{-}_{c}v_{p}]_{I}+\tilde{A}_{i}\chi^{-}_{k+1}(v_{p})-\tilde{v}_{p}\chi^{+}_{k+1+p}(A_{i}),$

where $a,b,c>0$. Since
$\partial\colon\mathfrak{l}/\bar{\mathfrak{g}}\to\operatorname{Hom}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})$
is injective, the above formula determines $\Psi^{+}_{k+1}$. ∎

Note that we know the finitude of the cohomology group:

######  Proposition 16.

There exists an integer $r_{0}$ such that

$H^{i}_{r}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})=0\quad\text{for
all }r>r_{0}\text{ and all }i.$

###### Proof.

The proof of Theorem 2.1 in [20] applies also to this case. ∎

Now we have

######  Theorem 4.

Let $\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$ be a morphism
of type $(\mathfrak{g}_{-},\mathcal{L})$. Then we can construct the series of
principal fibre bundles $Q(k)\to B(k-1)$ with $\mathfrak{l}$-valued 1-forms
$\omega(k)$ and the structure functions $\chi(k)^{(k)}\colon B(k)\to W^{(k)}$
for $k=0,1,2,\dots$. And $\chi(k_{0})^{(k_{0})}$ fulfills the complete set of
invariants of $\varphi$, where $k_{0}$ is an integer such that
$H^{1}_{k}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})=0$ for all
$k>k_{0}$.

Let $G_{-}$ (resp. $L_{-}$) be the Lie group with the Lie algebra
$\mathfrak{g}_{-}$ (resp. $\mathfrak{l}_{-}$). Then we have local embeddings
uniquely defined up to $\mathcal{L}$-equivalence:

$\varphi_{\text{model}}\colon G_{-}\to L_{-}\to(N,\mathfrak{f}_{N}),$

which we call _the standard embedding_.

######  Corollary.

Let $\varphi$, $\chi(k_{0})^{(k_{0})}$ be as in the preceding theorem. If
$\chi(k_{0})^{(k_{0})}=0$ identically, in particular, if
$H^{1}_{r}(\mathfrak{g}_{-},\mathfrak{l}/\bar{\mathfrak{g}})=0$ for $r>0$,
then $\varphi$ is formally $\mathcal{L}$-equivalent to the standard embedding
at every point of $M$.

In fact, if the hypothesis of the corollary holds, then the differential
equation for an $\mathcal{L}$-equivalence between $\varphi$ and
$\varphi_{\text{model}}$ is proved to be formally integrable and involutive,
and therefore has a formal solution. Moreover, it has a local analytic
solution in the analytic category. If $\mathfrak{l}$ is finite-dimensional,
then the differential equation is of finite type and has a local smooth
solution even in the $C^{\infty}$-category.

###  6.2. Examples

######  Example 7.

As an example, consider the embeddings
$\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$, where
$(N,\mathfrak{f}_{N})$ is a contact manifold of dimension $2n+1$. In other
words, $(N,\mathfrak{f}_{N})$ is a filtered manifold of constant type
$\mathfrak{m}=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}=\mathfrak{n}_{2n+1}$,
where $\mathfrak{n}_{2n+1}$ is a $(2n+1)$-dimensional Heisenberg Lie algebra.
Note that the Lie bracket $\wedge^{2}\mathfrak{m}_{-1}\to\mathfrak{m}_{-2}$
defines a symplectic form on $\mathfrak{m}_{-1}$ up to a non-zero scale.

In this case, $\mathfrak{l}$ is the (infinite-dimensional) graded Lie algebra
associated with the filtered Lie algebra of all contact vector fields on $N$.
Let $(x_{i},y,z_{i})$ be a local coordinate system on $N$ such that the
contact distribution $\mathfrak{f}_{N}^{-1}$ is defined as an annihilator of
the 1-form

$\theta=dy-z_{1}\,dx_{1}-\dots-z_{n}\,dx_{n}.$

Then it is well-known that all contact vector fields $X_{f}$ are parametrized
by a single function $f$ on $N$ such that

$\theta(X_{f})=f,\quad(L_{X_{f}}\theta)\wedge\theta=0.$

The Lie bracket of contact vector fields induces the bracket $\\{f,g\\}$ of
functions $f,g$ on $N$ such that

$X_{\\{f,g\\}}=[X_{f},X_{g}].$

Explicitly, we have:

$\\{f,g\\}=f\frac{\partial g}{\partial y}-g\frac{\partial f}{\partial
y}+\sum_{i=1}^{n}\left(\frac{df}{dx_{i}}\frac{\partial g}{\partial
z_{i}}-\frac{dg}{dx_{i}}\frac{\partial f}{\partial z_{i}}\right),$

where $\frac{d}{dx_{i}}=\frac{\partial}{\partial
x_{i}}+z_{i}\frac{\partial}{\partial y}$ for all $i=1,\dots,n$.

The Lie algebra $\mathfrak{l}$ can be described as a set of all contact vector
fields $X_{f}$, where $f$ runs through all polynomials in $(x_{i},y,z_{i})$,
$1\leq i\leq n$. In the following, we identify $\mathfrak{l}$ with the space
of such polynomials equipped with the above Lie bracket.

The Lie algebra $\mathfrak{m}$ can be identified with $\mathfrak{l}_{-}$,
which corresponds to the subspace

$\langle 1,x_{i},z_{i}\mid 1\leq i\leq n\rangle.$

Let us classify all graded subalgebras $\mathfrak{g}_{-}$ of $\mathfrak{m}$ up
to $\operatorname{Aut}_{0}(\mathfrak{m})$.

$\bullet$ Case 1: $\mathfrak{g}_{-}=\mathfrak{g}_{-1}\subset\mathfrak{m}$.
Then $\mathfrak{g}_{-}$ is necessarily commutative and thus, is an isotropic
subspace of $\mathfrak{m}_{-1}$. As the symplectic group acts transitively on
isotropic subspaces of the same dimension in $\mathfrak{m}_{-1}$, we see that
the $\mathfrak{g}_{-}$ is uniquely determined up to
$\operatorname{Aut}_{0}(\mathfrak{m})$ (=the grading preserving group of
automorphisms of $\mathfrak{m}$) by its dimension.

$\bullet$ Case 2: $\mathfrak{g}_{-}$ is 2-graded. Then we have
$\mathfrak{g}_{-2}=\mathfrak{m}_{-2}$. Assuming that $\mathfrak{g}_{-}$ is
generated by $\mathfrak{g}_{-1}$, we get that $\mathfrak{g}_{-1}$ can be an
arbitrary non-isotropic subspace of $\mathfrak{m}_{-1}$. Again, modulo the
action of $\operatorname{Aut}_{0}(\mathfrak{m})$ such subspaces are uniquely
determined by its dimension and the dimension of the kernel of the restriction
of the symplectic form to $\mathfrak{g}_{-1}$.

Let us compute $\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})$ in cases,
when $\mathfrak{g}_{-}$ is the maximal isotropic in $\mathfrak{m}_{-1}$ or
when $\mathfrak{g}_{-}$ is two-graded, and the restriction of the symplectic
form to $\mathfrak{g}_{-1}$ is non-degenerate.

$\bullet$ Case 1: $\mathfrak{g}_{-}$ is a maximal isotropic subspace in
$\mathfrak{l}_{-1}$. Then up to $\operatorname{Aut}_{0}(\mathfrak{m}),$ we can
assume that

$\mathfrak{g}_{-}=\langle z_{1},\dots,z_{n}\rangle.$

Let $\mathfrak{g}=\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})$. Then it
is easy to see that:

$\mathfrak{g}_{0}=\langle y,x_{i}z_{j},z_{i}z_{j}\mid 1\leq i,j\leq n\rangle.$

Further a simple computation shows that

$\mathfrak{g}=I(y,z_{1},\dots,z_{n},),$

the ideal generated by $y$ and $z_{1},\dots,z_{n}$. In particular, the
subspace $\mathfrak{g}^{\prime}$ consisting of of all polynomials in variables
$x_{i}$, $1\leq i\leq n$, forms a complementary linear subspace to
$\mathfrak{g}$ in $\mathfrak{l}$. Moreover, it is invariant with the action of
$\mathfrak{g}_{-}$.

Since $\\{x_{i},z_{j}\\}=\delta_{ij}$ for all $1\leq i,j\leq n$, we see that
the cohomology complex $C(\mathfrak{g}_{-},\mathfrak{g}^{\prime})$ coincides
with the _canonical Koszul complex_ , which is known to be acyclic. In
particular, we get that $H^{1}(\mathfrak{g}_{-},\mathfrak{g}^{\prime})=0$.

$\bullet$ Case 2: $\mathfrak{g}_{-}$ is 2-graded and $\mathfrak{g}_{-1}$ is
non-degenerate. Let $\dim\mathfrak{g}_{-1}=2m$. Then up to
$\operatorname{Aut}_{0}(\mathfrak{m}),$ we can assume that

$\mathfrak{g}_{-}=\langle 1,x_{i},z_{i}\mid 1\leq i\leq m\rangle.$

Computing $\mathfrak{g}_{0}$ we get

$\mathfrak{g}_{0}=\langle y,\quad x_{i}x_{j},x_{i}z_{j},z_{i}z_{j},\quad
x_{a}x_{b},x_{a}z_{b},z_{a}z_{b}\rangle,$

where $1\leq i,k\leq m$; $m+1\leq a,b\leq n$.

Further computations show that a polynomial $f$ belongs to $\mathfrak{g}$ if
and only if it has no terms linear in $(x_{a},z_{a})$, $m+1\leq a\leq n$. In
particular, as complementary linear subspace $\mathfrak{g}^{\prime},$ we can
choose

$\mathfrak{g}^{\prime}=\langle
x_{a}P_{a}(x_{i},y,z_{i}),z_{a}Q_{a}(x_{i},y,z_{i})\rangle,$

$1\leq i,k\leq m$; $m+1\leq a,b\leq n$, where $P_{a}$, $Q_{a}$, $m+1\leq a\leq
n$, are arbitrary polynomials in $(x_{i},y,z_{i})$, $1\leq i\leq m$.

Note that this space is invariant with respect to $\mathfrak{g}_{-}$ and
splits into the direct sum of $2(n-m)$ copies of submodules, which are
parametrized by a single polynomial $P(x_{i},y,z_{i})$ each. In its turn, each
such submodule is isomorphic to the graded Lie algebra $\mathfrak{l}^{\prime}$
associated to the filtered Lie algebra of all contact vector fields on the
$(2m+1)$-dimensional contact manifold. Since such Lie algebra coincides with
the Tanaka prolongation of $\mathfrak{g}_{-}$, we get that
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{l}^{\prime})=0$ and hence
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{g}^{\prime})=0$.

Thus, Corollary to Theorem 4 can be applied in both cases.

######  Example 8.

Consider now the extrinsic geometry of scalar 2nd order ODEs under contact and
point transformations.

We can consider a scalar 2nd order ODE as an embedding
$\varphi\colon(M,\mathfrak{f}_{M})\to(N,\mathfrak{f}_{N})$, where
$(N,\mathfrak{f}_{N})$ is a $4$-dimensional jet space
$J^{2}(\mathbb{R},\mathbb{R})$ with the canonical jet space filtration
$\mathfrak{f}_{N}$:

$\mathfrak{f}_{N}^{-1}\subset f_{N}^{-2}\subset\mathfrak{f}_{N}^{-3}=TN,$

and $M$ is a $3$-dimensional manifold (a 2nd order ODE) such that $\varphi(M)$
is transversal to the projection $\pi\colon J^{2}(\mathbb{R},\mathbb{R})\to
J^{1}(\mathbb{R},\mathbb{R})$ at each point $x\in\varphi(M)$. The filtration
$\mathfrak{f}_{M}$ is induced by the filtration $\mathfrak{f}_{N}$:

$\mathfrak{f}_{M}^{i}=\mathfrak{f}_{N}^{i}\cap TM\quad\text{for all
}i\in\mathbb{Z}.$

Let $\mathfrak{m}$ be the symbol algebra of $(N,\mathfrak{f}_{N})$. It is
well-known that
$\mathfrak{m}=\mathfrak{m}_{-1}\oplus\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-3}$
such that:

$\displaystyle\mathfrak{m}_{-1}$ $\displaystyle=\langle E_{2},X\rangle,$
$\displaystyle\mathfrak{m}_{-2}$ $\displaystyle=\langle E_{1}\rangle,$
$\displaystyle\mathfrak{m}_{-3}$ $\displaystyle=\langle E_{0}\rangle,$
$\displaystyle\text{where }[X,E_{i}]=E_{i-1},[E_{i},E_{j}]=0.$

Note that the fibres of the projection $\pi$ are tangent to the characteristic
direction of $\mathfrak{f}_{N}^{-2}$, which lies inside
$\mathfrak{f}_{N}^{-1}$ and corresponds to the subspace $\langle E_{2}\rangle$
in $\mathfrak{m}_{-1}$.

Since $M$ is transversal to this direction, the space $\varphi_{*}(T_{x}M)$ is
a $3$-dimensional vector space complementary to it, and
$\operatorname{gr}\varphi_{*}(T_{x}M)=\operatorname{gr}(\mathfrak{f}_{M})_{\varphi(x)}$
is a $3$-dimensional graded subalgebra of $\mathfrak{m}$ complementary to
$\langle E_{2}\rangle$. It is easy to see that all such subspaces are
equivalent to the fixed subspace:

$\mathfrak{g}_{-}=\langle X,E_{1},E_{0}\rangle$

under the action of $\operatorname{Aut}_{0}(\mathfrak{m})$.

Consider now the extrinsic invariants of the embedding $\varphi$ under contact
transformations. Let $\mathfrak{l}=\mathfrak{l}_{\text{cont}}$ be is the
(infinite-dimensional) graded Lie algebra associated with the filtered Lie
algebra of all vector fields on $N$ preserving the filtration
$\mathfrak{f}_{N}$. Due to Lie’s theorem all such vector fields are the lifts
of the contact vector fields on $J^{1}(\mathbb{R},\mathbb{R})$.

As in the previous example, the Lie algebra $\mathfrak{l}$ can be described as
a set of all contact vector fields $X_{f}$, where $f$ runs through all
polynomials in $(x,y,z)$. The grading of $\mathfrak{l}$ is defined as follows.
First, define weighted degree on $\mathbb{R}[x,y,z]$ by setting $\deg x=1$,
$\deg y=3$, $\deg z=2$. Then for any weighted homogeneous polynomial
$f\in\mathbb{R}[x,y,z]$ we define:

$\deg X_{f}=\deg f-3.$

In the following we identify $\mathfrak{l}$ with the space of such polynomials
equipped with the above Lie bracket.

######  Remark 4.

This construction easily generalizes to the description of the graded Lie
algebras associated with infinite-dimensional Lie algebra of vector fields
preserving the contact distribution on $J^{k}(\mathbb{R},\mathbb{R})$ for any
$k\geq 1$. Namely, in the case the (abstract) Lie algebra is still isomorphic
to $\mathfrak{l}$, but the grading is defined differently according to the
rules: $\deg x=1$, $\deg y=k+1$, $\deg z=k$, and $\deg X_{f}=\deg f-k-1$.

In this notation, the Lie algebra $\mathfrak{m}=\mathfrak{l}_{-}$ has the
form:

$\langle 1,x,x^{2},z\rangle.$

The subalgebra $\mathfrak{g}_{-}$ then corresponds to the subspace

$\mathfrak{g}_{-}=\langle 1,x,z\rangle.$

Let us compute
$\mathfrak{g}=\operatorname{Prol}(\mathfrak{g}_{-},\mathfrak{l})$. It is easy
to see that:

$\displaystyle\mathfrak{l}_{0}$ $\displaystyle=\langle y,x^{3},xz\rangle,$
$\displaystyle\mathfrak{g}_{0}$ $\displaystyle=\langle y,xz\rangle,$
$\displaystyle\mathfrak{l}_{1}$ $\displaystyle=\langle
xy,x^{4},x^{2}z,z^{2}\rangle,$ $\displaystyle\mathfrak{g}_{1}$
$\displaystyle=\langle x(y-xz),z^{2}\rangle.$

Further simple computations show that

$\mathfrak{g}=\\{P(z,y-xz)+Q(z,y-xz)x\\},$

where $P,Q$ are two arbitrary polynomials in 2 variables. Note that
$\mathfrak{g}$ coincides with all contact polynomial symmetries of the trivial
differential equation $y^{\prime\prime}=0$, or, in other words, with all
polynomial vector fields on $J^{1}(\mathbb{R},\mathbb{R})$ which preserve the
$1$-dimensional vector distribution $\langle d/dx\rangle$.

Let us show that $H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{l}/\mathfrak{g})=0$.
The short exact sequence

$0\to\mathfrak{g}\to\mathfrak{l}\to\mathfrak{l}/\mathfrak{g}\to 0$

induces the long exact sequence

(11) $\cdots\to H^{1}(\mathfrak{g}_{-},\mathfrak{l})\to
H^{1}(\mathfrak{g}_{-},\mathfrak{l}/\mathfrak{g})\to
H^{2}(\mathfrak{g}_{-},\mathfrak{g})\to\dots.$

So, it is sufficient to prove that
$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{l})=0$ and
$H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})=0$. This is done in two steps below.

Step 1. Denote by $\mathfrak{g}_{-}[-1]\subset\mathfrak{l}[-1]$ the same
abstract Lie algebras, but with the grading induced from
$J^{1}(\mathbb{R},\mathbb{R})$, that is by assuming that $\deg x=1$, $\deg
y=2$, $\deg z=1$ and $\deg X_{f}=\deg f-2$. Then it is well known that
$\mathfrak{l}[-1]$ is the Tanaka prolongation of $\mathfrak{g}_{-}[-1]$ and
hence $H^{1}_{r}(\mathfrak{g}_{-}[-1],\mathfrak{l}[-1])=0$ for all $r\geq 0$.
We note that $H^{1}(\mathfrak{g}_{-}[-1],\mathfrak{l}[-1])$ may have non-zero
components in negative degree. But it is easy to check that the negative part
of
$C^{1}(\mathfrak{g}_{-}[-1],\mathfrak{l}[-1])=\operatorname{Hom}(\mathfrak{g}_{-}[-1],\mathfrak{l}[-1])$
belongs to the negative part of $C^{1}(\mathfrak{g}_{-},\mathfrak{l})$. Hence,
we also get $H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{l})=0$.

Step 2. Let us prove that $H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})=0$. The
Legendre transform

$(x,y,z)\mapsto(-z,y-xz,x)$

induces an automorphism of $\mathfrak{l}$ that maps this vector distribution
to $\langle\partial/\partial z\rangle$ and, hence, maps the subalgebra
$\mathfrak{g}$ to the subalgebra of all polynomial vector fields on the plane
$(x,y)$ lifted to $J^{1}(\mathbb{R},\mathbb{R})$.

Thus, computing $H(\mathfrak{g}_{-},\mathfrak{g})$ we can identify
$\mathfrak{g}$ with the Lie algebra of all polynomial vector fields on the
plane with the grading induced from the conditions $\deg x=2,\deg y=3$. In
particular, under this identification we have:

$\mathfrak{g}_{-3}=\langle\partial/\partial
y\rangle,\quad\mathfrak{g}_{-2}=\langle\partial/\partial
x\rangle,\quad\mathfrak{g}_{-1}=\langle x\partial/\partial y\rangle.$

We note that $V=\langle\partial/\partial x,\partial/\partial y\rangle$ forms
an ideal in $\mathfrak{g}_{-}$. Moreover, the cohomology complex for the
$V$-module $\mathfrak{g}$ is exactly the standard Koszul cohomology complex,
which is known to be acyclic. Thus, we have $H^{q}(V,\mathfrak{g})=0$ for all
$q>0$ and $H^{0}(V,\mathfrak{g})=V$.

Now use Serre–Hochschild spectral sequence [9] associated with the ideal
$V\subset\mathfrak{g}_{-}$. Its second term $E_{2}^{p,q}$ is equal to:

$E_{2}^{p,q}=H^{p}(\mathfrak{g}_{-}/V,H^{q}(V,\mathfrak{g})).$

Since we want to compute $H^{2}(\mathfrak{g}_{-},\mathfrak{g})$, we can assume
that $p+q=2$. If $q>0$, then by the above $H^{q}(V,\mathfrak{g})=0$. Finally,
if $q=0$ and $p=2$, then

$H^{2}(\mathfrak{g}_{-}/V,H^{0}(V,\mathfrak{g}))=0,$

since $\mathfrak{g}_{-}/V$ is $1$-dimensional. This proves that
$H^{2}(\mathfrak{g}_{-},\mathfrak{g})=0$. In particular, this implies that an
arbitrary 2nd order ODE is contactly equivalent to the trivial equation
$y^{\prime\prime}=0$.

Consider now the case of the pseudogroup of point transformations on the
plane. Let $\mathfrak{l}=\mathfrak{l}_{\textrm{point}}$ be an infinite-
dimensional graded Lie algebra associated with the (filtered) Lie algebra of
all infinitesimal point transformations acting on
$J^{2}(\mathbb{R},\mathbb{R})$. It is well-known that
$\mathfrak{l}_{\textrm{point}}$ can be identified with the subalgebra of
$\mathfrak{l}_{\textrm{cont}}$ consisting of all vector fields $X_{f}$, where
$f$ is a polynomial in $x,y,z$ linear in $z$. We note that the grading of
$\mathfrak{l}$ is induced from its prolongation to
$J^{2}(\mathbb{R},\mathbb{R})$ and the standard filtration on
$J^{2}(\mathbb{R},\mathbb{R})$ (see above), which is different from the more
conventional grading on the Lie algebra of polynomial vector fields on the
place.

We have the same $\mathfrak{g}_{-}\subset\mathfrak{l}_{-}$ as in the contact
case. But the prolongation is now different. Similar to the contact case, it
is easy to show by direct computation that the prolongation of
$\mathfrak{g}_{-}$ in $\mathfrak{l}_{\textrm{point}}$ coincides with the point
symmetry algebra of the trivial equation $y^{\prime\prime}=0$, which is
exactly the Lie algebra $\mathfrak{sl}(3,\mathbb{R})$ of all projective vector
fields on the plane:

$\displaystyle\mathfrak{g}_{-3}$ $\displaystyle=\langle\partial/\partial
y\rangle,$ $\displaystyle\mathfrak{g}_{-2}$
$\displaystyle=\langle\partial/\partial x\rangle,$
$\displaystyle\mathfrak{g}_{-1}$ $\displaystyle=\langle x\partial/\partial
y\rangle,$ $\displaystyle\mathfrak{g}_{0}$ $\displaystyle=\langle
x\partial/\partial x,\quad y\partial/\partial y\rangle,$
$\displaystyle\mathfrak{g}_{1}$ $\displaystyle=\langle y\partial/\partial
x\rangle,$ $\displaystyle\mathfrak{g}_{2}$ $\displaystyle=\langle xU\rangle,$
$\displaystyle\mathfrak{g}_{3}$ $\displaystyle=\langle yU\rangle,$

where $U=x\partial/\partial x+y\partial/\partial y$.

As above, we can compute
$H_{+}^{1}(\mathfrak{g}_{-},\mathfrak{l}/\mathfrak{g})$ using the long exact
sequence (11). Next, we can prove that
$H^{k}_{+}(\mathfrak{g}_{-},\mathfrak{l})=0$ for $k=1$ and $k=2$ by means of
the Serre–Hochschild spectral sequence as in Step 2 above. Then (11) implies
the isomorphism

$H^{1}_{+}(\mathfrak{g}_{-},\mathfrak{l}/\mathfrak{g})\cong
H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g}).$

We note that the cohomology space on the right (with a different grading)
corresponds to the invariants of the parabolic geometry associated with a
scalar 2nd order ODE [2, 35]. So we see that the extrinsic geometry of the 2nd
order ODE under point transformations agrees the parabolic geometry associated
with this equation.

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Boris DOUBROV,
Belarusian State University,
Nezavisimosti ave. 4, 220030 Minsk, Belarus
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Yoshinori MACHIDA,
Shizuoka University,
Shizuoka 422-8529, Japan
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Tohru MORIMOTO,
Seki Kowa Institute of Mathematics, Yokkaichi University,
Yokkaichi 512-8045, Japan
Institut Kiyoshi Oka de Mathématiques, Nara Women’s University,
Nara 630-8506, Japan
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