The Riemannian curvature identities
on Almost Complex Calabi-Yau with torsion
6-manifold and generalized Ricci solitons

S. Ivanov1 and N. Stanchev2

1 University of Sofia, Faculty of Mathematics and Informatics,

blvd. James Bourchier 5, 1164, Sofia, Bulgaria

and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

2 University of Sofia, Faculty of Mathematics and Informatics,

blvd. James Bourchier 5, 1164, Sofia, Bulgaria

###### Abstract

Curvature properties of the torsion connection on an Almost complex CYT
6-manifold are investigated. It is observed that on a compact ACYT 6-manifold
the Nijenhuis tensor is parallel with respect to the torsion connection. It is
shown that a compact ACYT 6-manifold has closed torsion if and only if the
Ricci tensor of the torsion connection is equal to the covariant derivative of
the Lee form with respect to the torsion connection. It is proved that a
compact ACYT 6-manifold with closed torsion is Ricci flat if and only if
either the norm of the torsion is constant or the Riemannian scalar curvature
is constant. It is observed that any compact ACYT 6-manifold with closed
torsion 3-form is a generalized gradient Ricci soliton and this is equivalent
to a certain vector field to be parallel with respect to the torsion
connection. In particular, this vector field is an infinitesimal authomorphism
of the $SU(3)$ strucure.

It is also proved that on a compact ACYT 6-manifold the curvature of the
torsion connection $R\in S^{2}\Lambda^{2}$ and has vanishing Ricci tensor if
and only if the three-form torsion is parallel with respect to the Levi-Civita
and to the torsion connection simultaneously. In particular, the conditions
$R\in S^{2}\Lambda^{2},Ric=0$ are equivalent to the condition that the
curvature of the torsion connection satisfies the Riemannian first Bianchi
identity, i.e. it is Kähler-like. In this case the torsion 3-form is closed
and co-closed. In the complex case, it follows that the curvature of the
Strominger-Bismut connection on a six dimensional CYT space satisfies the
conditions $R\in S^{2}\Lambda^{2},Ric=0$ if and only if it is Kähler-like,
therefore flat, and has parallel torsion with respect to both the Levi-Civita
and Strominger-Bismut connections.

AMS MSC2010: 53C55, 53C21, 53C29, 53Z05

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Metric connection with skew-symmetric torsion and its curvature
    2. 2.2 Torsion connection preserving an Almost Hermitian structure
  3. 3 SU(3)-structures
  4. 4 Almost Calabi-Yau with torsion space in dimension 6
    1. 4.1 Example
  5. 5 Proof of Theorem 1.3and Theorem 1.4
  6. 6 ACYT with closed torsion and generalized Ricci solitons
    1. 6.1 Proof of Theorem 1.1
    2. 6.2 Generalized steady Ricci solitons

##  1 Introduction

Riemannian manifolds with metric connections having totally skew-symmetric
torsion and special holonomy received a lot of interest in mathematics and
theoretical physics mainly from supersymmetric string theories and
supergravity. The main reason becomes from the Hull-Strominger system which
describes the supersymmetric background in heterotic string theories [86, 53].
The number of preserved supersymmetries depends on the number of parallel
spinors with respect to a metric connection ${\nabla}$ with skew symmetric
torsion $T$. The torsion 3-form $T$ is identified with the 3-form field
strength in these theories.

The presence of a $\nabla$-parallel spinor leads to restriction of the
holonomy group $Hol(\nabla)$ of the torsion connection $\nabla$. Namely,
$Hol(\nabla)$ has to be contained in $SU(n),dim=2n$ [86, 44, 57, 58, 13, 7, 8,
49], the exceptional group $G_{2},dim=7$ [28, 45, 29], the Lie group
$Spin(7),dim=8$ [45, 56]. A detailed analysis of the possible geometries is
carried out in [44].

Complex Non-Kähler geometries appear in string compactifications and are
studied intensively for a long time [86, 39, 44, 45, 46, 39, 38, 7, 8].

Hermitian manifolds have widespread applications in both physics and
differential geometry in connection with solutions to the Hull-Strominger
system see [68, 30, 31, 24, 71, 72, 76, 74, 75, 77, 78, 22, 23, 20, 19, 34,
79] and references therein.

On a Hermitian manifold there exits a unique connection which preserves the
hermitian structure and has totally skew-symmetric torsion tensor. Its
existence and explicit expression first appeared in Strominger’s seminal paper
[86] in 1986 in connection with the heterotic supesymmetric string background,
where he called it the H-connection. Three years later, Bismut formally
discussed and used this connection in his local index theorem paper [11],
which leads to the name Bismut connection in literature. We call this
connection the Strominger-Bismut connection. Note that the connection also
appeared implicitly earlier ([88]) and in some literature it was also called
the KT connection (Kähler with torsion) or characteristic connection. When the
holonomy of the Strominger-Bismut connection is contained in $SU(n)$ one has
the notion of Calabi-Yau manifold whit torsion (CYT) spaces and these
manifolds are of great interest in string theories and in mathematics since
CYT appear in the Hull-Strominger system. Vanishing theorems for the Dolbeault
cohomology on compact Hermitian non-Kähler manifold were found in terms of the
Strominger-Bismut connection in [5, 58, 57, 90, 91].

If the torsion 3-form $T$ of $\nabla$ is closed, $dT=0$ which, in Hermitian
manifold, is equivalent to the condition $\partial\bar{\partial}F=0$, where
$F$ is the fundamental 2-form, the Hermitian metric g is called SKT (strong
Kähler with torsion) [51] or pluriclosed. The SKT (pluriclosed) metrics has
found many applications in both physics, see eg [36, 52, 86, 47, 48, 21] and
geometry, see eg [61, 57, 58, 25, 30, 31, 24, 26, 83, 82] and references
therein. For example in type II string theories, the three form $T$ is
identified with the 3-form field strength and it is required by construction
to satisfy $dT=0$. In a series of papers Streets and Tian [83, 84, 85]
introduced a hermitian Ricci flow under which the pluriclosed or equivalently
strong KT structure is preserved, show the relationship of pluriclosed flow to
type II string backgrounds and find that pluriclosed flow preserves N=(2,2)
supersymmetry (generalized Kahler geometry). Compact pluriclosed (strong) CYT
spaces are closely connected with the pluriclosed flow and the equations of
motion in compactifications of type II supergravity due to the recent work of
Garcia-Fernandez, Jordan and Streets [33]. Moreover, a global existence and
convergence of pluriclosed flow to a Stromonger-Bismut-flat metric on complex
manifolds admitting Strominger-Bismut-flat metrics is also shown in [33].

Generalizations of the pluriclosed condition $\partial\bar{\partial}F=0$ on 2n
dimensional Hermitian manifolds in the form
$F^{\ell}\wedge\partial\bar{\partial}F^{k}=0~{},~{}~{}~{}1\leq k+\ell\leq
n-1~{}$ has been investigated in [27, 80, 32, 59] etc.

Hermitian metrics with the Strominger-Bismut connection being Kähler-like,
namely, its curvature satisfies the Riemannian first Bianchi identity ((2.10)
below) have been studied in [6], investigating this property on 6-dimensional
solvmanifolds with holomorphically trivial canonical bundle and it is
conjectured by Angela-Otal-Ugarte-Villakampa in [6] that such metrics should
be SKT (pluriclosed).

A support to that can be derived from [Theorem 5][87] where the simply
connected hermitian manifolds with flat Strominger-Bismut connection are
classified and it easy follows that it is SKT (pluriclosed) with parallel
3-form torsion [6]. The latter conclusion also follows from the Cartan-
Schouten theorem [12], (see also [Theorem 2.2][3]).

The conjecture by Angela-Otal-Ugarte-Villakampa was proved in [92]. It is also
shown in [92] that in this case the torsion is closed and ${\nabla}$-parallel,
$dT=\nabla T=0$.

Hermitian nilmanifolds whose Strominger-Bismut connection is Kähler-like are
classified in [93]. A classification of compact non-Kähler Hermitian manifolds
with Kähler-like Strominger-Bismut connection in complex dimension 3 and those
with degenerate torsion in higher dimensions is given in [90]. For complex
dimension three [90, Theorem 5] shows that if the space is CYT and the
Strominger-Bismut connection is Kähler-like then the Strominger-Bismut
connection is flat and it is conjectured that this is true in higher
dimensions provided the space is compact, [90, Conjecture 2]. This conjecture
was confirmed very recently in [95].

There is also an interesting geometric explanation of the curvature of the
Strominger-Bismut connection in the frame work of generalized Ricci flow
developed by Garcia-Fernandez and Streets [35] (see also the references
therein).

Some types of non-complex 6-manifold have been also invented in the string
theory due to the mirror symmetry and T-duality [62, 42, 43, 13, 14, 65, 64,
50].

Almost Hermitian manifolds with totally skew-symmetric Nijenhuis tensor arise
as target spaces of a class of (2,0)-supersymmetric two-dimensional sigma
models [73]. For the consistency of the theory, the Nijenhuis tensor has to be
parallel with respect to the torsion connection with holonomy contained in
$SU(n)$. The known models are those on group manifolds. A 6-dimensional
nilmanifold, which is neither complex nor nearly Kaehler, is presented in
[54].

In general, an almost Hermitian manifold does not admits a metric connection
preserving the almost Hermitian structure and having torsion 3-form. It is
shown in [28, Theorem 10.1] that this is equivalent to the condition that the
Nijenhuis tensor is totally skew-symmetric, i.e. this is the class $G_{1}$ in
Gray-Hervella classification [41]. In this case the connection is unique. This
class contains the hermitian manifolds with Strominger-Bismut connection as
well as the nearly Kaehler spaces. In the nearly Kaehler case the torsion
connection ${\nabla}$ is the characteristic connection considered by Gray,
[40] and the torsion $T$ and the Nijenhuis tensor are ${\nabla}$-parallel,
${\nabla}T={\nabla}N=0$ (see e.g. [63, 9]). If the holonomy of the torsion
connection is contained in $SU(n)$, that is, the Ricci 2-form $\rho$ of
${\nabla}$ representing the first Chern class is identically zero, then we
have the notion of _Almost Calabi-Yau manifold with torsion (briefly ACYT)_.

As it was pointed out in [86], the existence of a parallel spinor with respect
to a metric connection ${\nabla}$ with torsion 3-form, in dimension six, leads
to the restriction that its holonomy group has to be contained in $SU(3)$.
This means one has to consider an $SU(3)$-structure, i.e. an almost hermitian
manifold $(M^{6},g,J)$ with topologically trivial canonical bundle trivialized
by a (3,0) with respect to $J$ form $\Psi=\Psi^{+}+\sqrt{-1}\Psi^{-}$ endowed
with a metric connection preserving the $SU(3)$ structure with totally skew-
symmetric torsion.

Special attention is paid when the torsion 3-form is closed, $dT=0$. For
example in type II string theory, $T$ is identified with the 3-form field
strength. This is required by construction to satisfy $dT=0$ (see e.g. [45,
44]). More generally, the geometry of a torsion connection with closed torsion
form, $dT=0$ appears in the frame work of the generalized Ricci flow and the
generalized (gradient) Ricci solitons developed by Garcia-Fernandez and
Streets [35] (see the references therein). Concerning the CYT spaces, it is
shown in [35, Proposition 8.14], (see also [33, Proposition 2.6] that any
compact pluriclosed CYT space is automatically a steady generalized gradient
Ricci soliton [35, Proposition 8.14], (see also [33, Proposition 2.6].

Our main purposes are to investigate the curvature properties of the torsion
connection on an ACYT 6-manifold.

We observe in Theorem 6.1 that an ACYT 6-manifold has closed torsion, $dT=0$
if and only if the Ricci tensor $Ric$ of the torsion connection ${\nabla}$ is
determined by the covariant derivative of the Lee form with the formula
$Ric=-{\nabla}\theta$ and the Nijenhuis tensor $N$ is ${\nabla}$-parallel.
These conditions are also equaivalent to the conditions $Ric=-{\nabla}\theta$
and $d\theta\in su(3)\cong\Lambda^{2}_{8}$. In the compact case, we obtain in
Theorem 4.4 that on any compact 6-dimensional ACYT the Nijenhuis tensor is
${\nabla}$ parallel and deduce in Corollary 6.2 that a compact ACYT 6-manifold
has closed torsion, $dT=0$, exactly when $Ric=-{\nabla}\theta$. This helps to
prove our first main result,

######  Theorem 1.1.

Let $(M,g,J,\Psi)$ be a 6-dimensional compact ACYT space with closed torsion,
$dT=0$.

The following conditions are equivalent:

  * a).

The torsion connection is Ricci flat, $Ric=0$;

  * b).

The norm of the torsion is constant, $d||T||^{2}=0$;

  * c).

The Lee form is $\nabla$-parallel, ${\nabla}\theta=0$;

  * d).

The torsion connection has vanishing scalar curvature, $Scal=0$;

  * e).

The Riemannian scalar curvature is constant, $Scal^{g}=const.$;

  * f).

The Lee form is co-closed, $\delta\theta=0$.

In each of the six cases above the torsion is coclosed and therefore it is
harmonic 3-form.

In the complex case, $N=0$, we have

######  Corollary 1.2.

A compact pluriclosed CYT 6-manifold is Strominger-Bismut Ricci flat if and
only if either the torsion has constant norm or the Riemannian scalar
curvature is constant.

In particular, the torsion is harmonic.

We show in Theorem 6.5 that any compact ACYT 6-manifold with closed torsion
3-form is a steady generalized gradient Ricci soliton and this condition is
equivalent to a certain vector field to be parallel with respect to the
torsion connection. We also find out that this vector field preserves the
$SU(3)$ strucure, thus extending the above mentioned result in [33] for
compact pluriclosed CYT space to compact ACYT space with closed torsion in
dimension six.

Studying the curvature properties of the torsion connection on an ACYT
6-manifold we find necessary conditions the torsion 3-form to be closed and
harmonic,

######  Theorem 1.3.

Let $(M,g,J,\Psi)$ be a 6-dimensional ACYT space. The torsion connection
${\nabla}$ preserving the $SU(3)$ structure is Ricci flat, has curvature $R\in
S^{2}\Lambda^{2}$ and the Nijenhuis tensor is of constant norm,

$R(X,Y,Z,V)=R(Z,V,X,Y),\qquad Ric(X,Y)=0,\qquad||N||^{2}=const.$ (1.1)

if and only if the torsion $T$ is ${\nabla^{g}}$ and ${\nabla}$-parallel,
${\nabla^{g}}T={\nabla}T=0.$

In particular the torsion 3-form is harmonic and he Nijenhuis tensor, the
exterior derivative $dF$ of the Kaehler 2-form are ${\nabla}$-parallel,

${\nabla}N={\nabla}dF=0.$ (1.2)

The curvature of the torsion connection obeys the Riemannian first Bianchi
identity (2.10) and vice-versa.

Taking into account Theorem 4.4 below, in the compact case, Theorem 1.3 can be
stated as follows

######  Theorem 1.4.

Let $(M,g,J,\Psi)$ be a compact 6-dimensional ACYT space.The torsion
connection ${\nabla}$ preserving the $SU(3)$ structure has curvature $R\in
S^{2}\Lambda^{2}$ with vanishing Ricci tensor

$R(X,Y,Z,V)=R(Z,V,X,Y),\qquad Ric(X,Y)=0.$ (1.3)

if and only if the torsion 3-form $T$ is ${\nabla^{g}}$ and
${\nabla}$-parallel, ${\nabla^{g}}T={\nabla}T=0.$

In particular the torsion 3-form is harmonic, $dT=\delta T=0$ and (1.2) holds.

The curvature of the torsion connection obeys the Riemannian first Bianchi
identity (2.10) and vice-versa.

We can reformulate the above result as follows

######  Corollary 1.5.

On a compact ACYT manifold in dimension 6 the next conditions are equivalent:

  * •

The torsion connection has curvature $R\in S^{2}\Lambda^{2}$ with vanishing
Ricci tensor;

  * •

The curvature of the torsion connection is Kähler like ;

  * •

The torsion is closed and parallel $dT={\nabla^{g}}T={\nabla}T=0$.

In the Hermitian case, $N=0$, the conclusion ${\nabla}T=0$ follows from the
assumptions of Theorem 1.3 due to [94, Theorem 1.1] in all dimensions.

We remark that, in general, metric connections with skew symmetric torsion $T$
are closely connected with the generalized Ricci flow. Namely, the fixed
points of the generalized Ricci flow are Ricci flat metric connections with
harmonic torsion 3-form, $Ric=dT=\delta T=0$, we refer to the recent book [35]
and the references given there for mathematical and physical motivation.

In this direction, our results show that a 6-dimensional ACYT manifold with
constant norm of the Nijenhuis tensor (in particular any compact ACYT
manifold) with Ricci flat torsion connection having curvature $R\in
S^{2}\Lambda^{2}$ is a fixed point of the corresponding generalized Ricci
flow. In particular, if the curvature of the torsion connection satisfies the
Riemannian first Bianchi identity then it is a fixed point of the generalized
Ricci flow. In the compact case, any ACYT 6-manifold with closed torsion and
either constant Riemannian scalar curvature or constant norm of the torsion is
a fixed point of the generalized Ricci flow.

Theorem 1.3 help to generalize [90, Theorem 5] as follows

######  Theorem 1.6.

Let $M^{6}$ be a CYT whose Strominger-Bismut connection ${\nabla}$ satisfies
(1.1). Then it is Strominger-Bismut Kähler-like and therefore is Strominger-
Bismut flat.

Note that Hermitian and almost Hermitian menifolds of type $G_{1}$ with
${\nabla}$\- parallel torsion 3-form are investigated in [4, 81], recently in
[18, 94, 70] and a large number of examples are given there. More generally,
Riemannian manifolds equiped with metric connection ${\nabla}$ with
${\nabla}$-parallel torsion 3-form are investigated in [2, 1, 16, 17].

It is conjectured very recently that if a complete Hermitian manifold has
Strominger-Bismut parallel curvature and torsion and zero all Ricci tensors
then it has to be Strominger-Bismut flat, [70, Conjecture 1.8]. A spesial case
with the additional conditions that the space is Kaehler-like and CYT is
proved in [70, Proposition 6.8].

In this direction, it is worth to mention [2, Theorem 4.1] which states that
an irreducible complete and symply connected Riemannian manifold of dimension
bigger or equal to $5$ with ${\nabla}$-parallel and closed torsion 3-form,
${\nabla}T=dT=0$, is a simple compact Lie group or its dual non-compact
symmetric space with biinvariant metric, and, in particular the torsion
connection is the flat Cartan connection.

######  Remark 1.7.

A combination of the above mentioned [92, Theorem 1] with [2, Theorem 4.1]
shows that an irreducible complete and symply connected Hermitian manifold of
dimension bigger or equal to 6 which is Kaehler-like should be a simple
compact Lie group with flat Strominger-Bismut connection. Moreover, due to
Theorem 1.4 and [2, Theorem 4.1], irreducible symply connected compact
6-dimensional ACYT space satisfying conditions (1.3) does not exists since
there are no simple compact Lie groups of dimension six.

We also note that the main result [92, Theorem 1] mentioned above combined
with [58, Corollary 4.7] yields in a straightforward way the next

######  Corollary 1.8.

Let $M$ be a 2n-dimensional compact Hermitian manifold and its Strominger-
Bismut connection is Kähler-like. If $M$ is conformally balanced (the Lee form
is an exact form) then the space is a Kähler manifold.

######  Convention 1.9.

Everywhere in the paper we will make no difference between tensors and the
corresponding forms via the metric as well as we will use Einstein summation
conventions ie repeated Latin indices are summed over.

##  2 Preliminaries

In this section we recall some known curvature properties of a metric
connection with skew symmetric torsion on Riemannian manifold as well as the
notions and existence of a metric linear connection preserving a given Almost
Hermitian structure and having skew symmetric torsion from [55, 28, 60].

###  2.1 Metric connection with skew-symmetric torsion and its curvature

On a Riemannian manifold $(M,g)$ of dimension $n$ any metric connection
${\nabla}$ with totally skew-symmetric torsion $T$ is connected with the Levi-
Civita connection ${\nabla}^{g}$ of the metric $g$ by

${\nabla}^{g}={\nabla}-\frac{1}{2}T.$ (2.4)

The exterior derivative $dT$ has the following expression (see e.g. [55, 58,
28])

$\begin{split}dT(X,Y,Z,V)=(\nabla_{X}T)(Y,Z,V)+(\nabla_{Y}T)(Z,X,V)+(\nabla_{Z}T)(X,Y,V)\\\
+2\sigma^{T}(X,Y,Z,V)-(\nabla_{V}T)(X,Y,Z),\end{split}$ (2.5)

where the 4-form $\sigma^{T}$, introduced in [28], is defined by
$\sigma^{T}(X,Y,Z,V)=\frac{1}{2}\sum_{j=1}^{n}(e_{j}\lrcorner
T)\wedge(e_{j}\lrcorner T)(X,Y,Z,V)$, $(e_{a}\lrcorner T)(X,Y)=T(e_{a},X,Y)$
is the interior multiplication and $\\{e_{1},\dots,e_{n}\\}$ is a basis. The
properties of the 4-form $\sigma^{T}$ are studied in detail in [2] where it is
shown that $\sigma^{T}$ measures the ‘degeneracy’ of the 3-form $T$.

The identity (2.4) yields

${\nabla^{g}}T={\nabla}T+\frac{1}{2}\sigma^{T}.$ (2.6)

For the curvature of ${\nabla}$ we use the convention
$R(X,Y)Z=[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z$ and
$R(X,Y,Z,V)=g(R(X,Y)Z,V)$. It has the well known properties

$R(X,Y,Z,V)=-R(Y,X,Z,V)=-R(X,Y,V,Z).$ (2.7)

The first Bianchi identity for $\nabla$ can be written in the form (see e.g.
[55, 58, 28])

$\begin{split}R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)\\\
=dT(X,Y,Z,V)-\sigma^{T}(X,Y,Z,V)+(\nabla_{V}T)(X,Y,Z).\end{split}$ (2.8)

It is proved in [28, p.307] that the curvature of a metric connection
${\nabla}$ with totally skew-symmetric torsion $T$ satisfies also the identity

$\begin{split}R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)-R(V,X,Y,Z)-R(V,Y,Z,X)-R(V,Z,X,Y)\\\
=\frac{3}{2}dT(X,Y,Z,V)-\sigma^{T}(X,Y,Z,V),\end{split}$

which combined with (2.8) yields

$\begin{split}R(V,X,Y,Z)+R(V,Y,Z,X)+R(V,Z,X,Y)=-\frac{1}{2}dT(X,Y,Z,V)+(\nabla_{V}T)(X,Y,Z)\end{split}$
(2.9)

######  Definition 2.1.

We say that the curvature $R$ satisfies the Riemannian first Bianchi identity
if

$R(X,Y,Z,V)+R(Y,Z,X,V)+R(Z,X,Y,V)=0.$ (2.10)

It is well known algebraic fact that (2.7) and (2.10) imply $R\in
S^{2}\Lambda^{2}$, i.e.

$R(X,Y,Z,V)=R(Z,V,X,Y)$ (2.11)

holds.

######  Remark 2.2.

Note that, in general, (2.7) and (2.11) do not imply (2.10).

It is known that a metric connection ${\nabla}$ with skew-symmetric torsion
$T$ has curvature $R\in S^{2}\Lambda^{2}$, i.e. it satifies (2.11) if and only
if the the covariant derivative of the torsion with respect to the torsion
connection ${\nabla}T$ is a 4-form.

######  Lemma 2.3.

[55, Lemma 3.4] The next equivalences hold for a metric connection with
torsion 3-form

$({\nabla}_{X}T)(Y,Z,V)=-({\nabla}_{Y}T)(X,Z,V)\Longleftrightarrow
R(X,Y,Z,V)=R(Z,V,X,Y))\Longleftrightarrow dT=4{\nabla^{g}}T.$ (2.12)

We need the very recent result from [60]

######  Theorem 2.4.

[60, Theorem 1.2] A metric connection ${\nabla}$ with torsion 3-form $T$
satisfies the Riemannian first Bianchi identity exactly when the next
identities hold

$dT=-2\nabla T=\frac{2}{3}\sigma^{T}.$

In this case, the torsion 3-form $T$ is parallel with respect to a metric
connection with torsion 3-form $\frac{1}{3}T$ [3] and therefore has a constant
norm, $||T||^{2}=const.$

The Ricci tensors and scalar curvatures of the connections ${\nabla^{g}}$ and
${\nabla}$ are related by [28, Section 2], (see also [35, Prop. 3.18])

$\begin{split}Ric^{g}(X,Y)=Ric(X,Y)+\frac{1}{2}(\delta
T)(X,Y)+\frac{1}{4}\sum_{i=1}^{n}(g(T(X,e_{i}),T(Y,e_{i});\\\
Scal^{g}=Scal+\frac{1}{4}||T||^{2},\qquad Ric(X,Y)-Ric(Y,X)=-(\delta
T)(X,Y),\end{split}$ (2.13)

where $\delta=(-1)^{np+n+1}*d*$ is the co-differential acting on $p$-forms and
$*$ is the Hodge star operator satisfying $*^{2}=(-1)^{p(n-p)}$.

###  2.2 Torsion connection preserving an Almost Hermitian structure

In this section we recall the notions and existence of a metric linear
connection preserving a given Almost Hermitian structure and having totally
skew-symmetric torsion from [28].

Let $(M,g,J)$ be an almost Hermitian 2n-dimensional manifold with Riemannian
metric $g$ and an almost complex structure $J$ satisfying

$g(JX,JY)=g(X,Y).$

The Nijenhuis tensor $N$, the Kähler form $F$ and the Lee form $\theta$ are
defined by

$N=[J.,J.]-[.,.]-J[J.,.]-[.,J.],\quad F=g(.,J.),\quad\theta(.)=\delta F(J.),$
(2.14)

respectively, where $\delta=-*d*$ is the co-differential on an even
dimensional manifold.

Consequently, the 1-form $J\theta$ defined by $J\theta(X)=-\theta(JX)$ is co-
closed due to $\delta(J\theta)=-\delta^{2}F=0$.

The Nijenhuis tensor of type (0,3) is defined by $N(X,Y,Z)=g(N(X,Y),Z)$.

In general, on an Almost hermitian manifold there not always exists a metric
connection with totally skew-symmetric torsion preserving the almost hermitian
structure. It is shown in [28, Theorem 10.1] that there exists a unique linear
connection ${\nabla}$ preserving an almost hermitian structure $(g,J)$,

${\nabla}g={\nabla}J=0$

and having totally skew-symmetric torsion $T$ if and only if the Nijenhuis
tensor $N$ is a 3-form. This is precisely the class $G_{1}$ in the Gray-
Hervella classification of almost Hermitian manifods [41] and includes
Hermitian manifolds, $N=0$ as well as nearly Kaehler spaces,
$({\nabla^{g}}_{X}J)X=0$.

The torsion $T$ is determined by [28, Theorem 10.1]

$T=JdF+N=-dF(J.,J.,J.)+N=-dF^{+}(J.,J.,J.)+\frac{1}{4}N,$ (2.15)

where $dF^{+}$ denotes the (1,2)+(2,1)-part of $dF$ with respect to $J$ and
satisfies the equality

$\begin{split}dF^{+}(X,Y,Z)=dF^{+}(JXJ,Y,Z)+dF^{+}(JX,Y,JZ)+dF^{+}(X,JY,JZ).\end{split}$
(2.16)

It follows from (2.16) that

$dF^{+}(Je_{k},Je_{i},e_{j})dF^{+}(e_{k},e_{i},e_{j})=\frac{1}{3}||dF^{+}||^{2}.$
(2.17)

The (3,0)+(0,3)-part $dF^{-}$ obeys the identity

$dF^{-}(JX,Y,Z)=dF^{-}(X,JY,Z)=dF^{-}(X,Y,JZ)$

and is determined completely by the Nijenhuis tensor [37]. If $N$ is a three
form then (see e.g.[37, 28])

$dF^{-}(X,Y,Z=\frac{3}{4}N(JX,Y,Z).$

The Lee form $\theta:=\delta F\circ J$ of the $G_{1}$\- manifold is given in
terms of the torsion $T$ as

$\theta(X)=-\frac{1}{2}T(JX,e_{i},Je_{i})=\frac{1}{2}g(T(JX),F)=\frac{1}{2}(F\lrcorner
T(JX))~{},~{}~{}~{}\theta_{i}={1\over 2}J^{k}{}_{i}T_{kj\ell}F_{j\ell}~{},$
(2.18)

where $\\{e_{1},\dots,e_{n},e_{n+1}=Je_{1},\dots,e_{2n}=Je_{n}\\}$ is an
orthonormal basis.

When the almost complex structure is integrable, $N=0$, the torsion connection
was defined and studied by Strominger [86] in connection with the heterotic
string background and was used by Bismut to prove local index theorem for the
Dolbeault operator on Hermitian non-Kähler manifold [11]. This formula was
applied also in string theory (see e.g. [8]). In the hermitian case the
torsion connection is also known as the Strominger-Bismut connection and has
attained a lot of considerations both in mathematics and physics (see the
Introduction).

In the nearly Kaehler case the torsion connection ${\nabla}$ is the
characteristic connection considered by Gray, [40] and the torsion $T$ and the
Nijenhuis tensor are ${\nabla}$-parallel, ${\nabla}T={\nabla}N=0$ (see e.g.
[63, 9]).

In general $G_{1}$ manifold, the holonomy group of ${\nabla}$ is contained in
$U(n)$. The reduction of the holonomy group of the torsion connection to a
subgroup of $SU(n)$ can be expressed in terms of its Ricci 2-form wich
represents the first Chern class, namelly

$\rho(X,Y)=\frac{1}{2}R(X,Y,e_{i},Je_{i})=0.$

We remark that the formula [58, (3.16)] holds in the general case of a
$G_{1}$-manifold (see also [28])

$\rho(X,Y)=Ric(X,JY)+({\nabla}_{X}\theta)(JY)+\frac{1}{4}dT(X,Y,e_{i},Je_{e}).$
(2.19)

Thus, on a $G_{1}$ manifold the restricted holonomy group of the torsion
connection is contained in $SU(n)$ if and only if the next condition holds
[28, Theorem 10.5]

$Ric(X,Y)+({\nabla}_{X}\theta)(Y)-\frac{1}{4}dT(X,JY,e_{i},Je_{e})=0.$ (2.20)

We calculate from (2.5) using (2.15) and (2.17)

$dT(e_{j},Je_{j},e_{i},Je_{i})=-8({\nabla}_{e_{i}}\theta)(e_{i})+8|\theta|^{2}-4T(e_{i},e_{j},e_{k})T(Je_{i},Je_{j},e_{k})\\\
=8\delta\theta+8||\theta||^{2}-\frac{4}{3}||dF^{+}||^{2}+\frac{1}{4}||N||^{2}=8\delta\theta+8|\theta|^{2}-\frac{4}{3}||T||^{2}+\frac{1}{3}||N||^{2}.$
(2.21)

######  Remark 2.5.

It follows from (2.21) that if the structure is balanced, $\theta=0$, and the
torsion is closed, $dT=0$ then $16|dF|^{2}=3|N|^{2}$. In the complex case,
$N=0$, this confirmes the old result first established in [5] that the space
must be Kaehler.

The trace in (2.20) together with (2.21) and (2.13) yield the following
formulas for the scalar curvature of torsion connection and the Riemannian
scalar curvature on an Almost complex CYT space

$\begin{split}Scal=3\delta\theta+2||\theta||^{2}-\frac{1}{3}||dF^{+}||^{2}+\frac{1}{16}||N||^{2};\qquad
Scal^{g}=3\delta\theta+2||\theta||^{2}-\frac{1}{3}||dF^{+}||^{2}+\frac{5}{64}||N||^{2}.\end{split}$
(2.22)

We obtain from (2.19), (2.22) and (2.21)

######  Theorem 2.6.

Let $(M,g,J)$ be a 2n-dimensional balanced ACYT space.

  * a)

If either the Riemannian scalar curvature or the scalar curvature of the
torsion connection vanishes, $s^{g}=0$ or $s=0$ then

$16||dF^{+}||^{2}=3||N||^{2}.$ (2.23)

  * b)

If the Ricci tensor of the torsion connection vanishes, $Ric=0$ then
$dT(X,Y,e_{i},Je_{i})=0.$

In particular, (2.23) holds.

Consequently, we have

######  Corollary 2.7.

On a balanced CYT manifold if either the Riemannian scalar curvature or the
scalar curvature of the Strominger-Bismut connection vanishes, $Scal^{g}=0$ or
$Scal=0$ then it must be a Kaehler Ricci flat, i.e. a Calabi-Yau space.

In particular, any balanced CYT space with vanishing Ricci tensor of the
Strominger-Bismut connection is Kaehler Ricci flat, i.e. it is a Calabi-Yau
space.

Following [6] we have

######  Definition 2.8 ([6]).

We say that a curvature tensor $R$ is Kähler like if it resembles the Kähler
curvature identites, i.e. in addition to (2.7) and (2.10) it satisfies

$\begin{split}R(X,Y,JZ,JV)=R(X,Y,Z,V);\end{split}$ (2.24)

Since the torsion connection on an almost Hermitian manifold with skew-
symmetric Nijenhuis tensor preserves the hermitian metric and the complex
structure, $\nabla g=\nabla J=0$ its curvature satisfies (2.7) and (2.24).
Clearly, the curvature of the torsion connection is of Kähler type exactly
when it satisfies the Riemannian first Bianchi identity (2.10). In this case
the Ricci tensor is symmetric, $J$-invariant and $Ric(X,JY)=\rho(X,Y)$.

##  3 SU(3)-structures

Let $(M,g,J)$ be an almost Hermitian 6-manifold with Riemannian metric $g$ and
an almost complex structure $J$, i.e. $(g,J)$ define an $U(3)$-structure.

An $SU(3)$-structure is determined by an additional non-degenerate (3,0)-form
$\Psi=\Psi^{+}+\sqrt{-1}\Psi^{-}$ of constant norm, or equivalently by a non-
trivial spinor. To be more explicit, we may choose a local orthonormal frame
$e_{1},\dots,e_{6}$, identifying it with the dual basis via the metric. Write
$e_{i_{1}i_{2}\dots i_{p}}$ for the monomial $e_{i_{1}}\wedge
e_{i_{2}}\wedge\dots\wedge e_{i_{p}}$. An $SU(3)$-structure is described
locally by

$\begin{split}\Psi=-(e_{1}+\sqrt{-1}e_{2})\wedge(e_{3}+\sqrt{-1}e_{4})\wedge(e_{5}+\sqrt{-1}e_{6}),\quad
F=-e_{12}-e_{34}-e_{56},\\\
\Psi^{+}=-e_{135}+e_{236}+e_{146}+e_{245},\quad\Psi^{-}=-e_{136}-e_{145}-e_{235}+e_{246}.\end{split}$
(3.25)

These forms are subject to the compatibility relations

$F\wedge\Psi^{\pm}=0,\qquad\Psi^{+}\wedge\Psi^{-}=-\frac{2}{3}F^{3};$

The subgroup of $SO(6)$ fixing the forms $F$ and $\Psi$ simultaneously is
$SU(3)$. The two forms $F$ and $\Psi$ determine the metric completely. The Lie
algebra of $SU(3)$ is denoted $su(3)$.

The failure of the holonomy group of the Levi-Civita connection ${\nabla}^{g}$
of the metric $g$ to be reduced to $SU(3)$ can be measured by the intrinsic
torsion $\tau$, which is identified with $dF,d{\Psi^{+}}$ and $d{\Psi^{-}}$
and can be decomposed into five classes [15], $\tau\in W_{1}\oplus\dots\oplus
W_{5}$. The intrinsic torsion of an $U(n)$\- structure belongs to the first
four components described by Gray-Hervella [41]. The five components of a
$SU(3)$-structure are first described by Chiossi-Salamon [15] (for
interpretation in physics see [13, 42, 43]) and are determined by
$dF,d\Psi^{+},d\Psi^{-}$ as well as $dF$ and $N$. We list below those of these
classes which we will use later.

$\tau\in W_{1}:$

The class of Nearly Kähler (weak holonomy) manifold defined by $dF$ to be
(3,0)+(0,3)-form.

$\tau\in W_{3}:$

The class of balanced hermitian manifold determined by the conditions
$N=\theta=0$.

$\tau\in W_{1}\oplus W_{3}\oplus W_{4}:$

The class called by Gray-Hervella $G_{1}$-manifolds determined by the
condition that the Nijenhuis tensor is totally skew-symmetric. This is the
precise class which we are interested in.

$\tau\in W_{1}\oplus W_{3}:$

The class of balanced $G_{1}$-manifolds determined by the condition that the
Nijenhuis tensor is totally skew-symmetric and the Lee form vanishes,
$N(X,Y,Z)=-N(X,Z,Y),\quad\theta=0$.

$\tau\in W_{3}\oplus W_{4}:$

The class of Hermitian manifolds, $N=0$.

If all five components are zero then we have a Ricci-flat Kähler (Calabi-Yau)
3-fold.

Let $(M,g,J,\Psi)$ be an $SU(3)$ manifold. Letting the four form

$\Phi=\frac{1}{2}F^{2}=\frac{1}{2}F\wedge F$

we have the relations

$\begin{split}*\Phi=-F,\quad*F=-\Phi;\qquad*\Psi^{+}=\Psi^{-},\quad*\Psi^{-}=-\Psi^{+}\\\
*(\alpha\wedge
F)=-\alpha\lrcorner\Phi,\quad\alpha\in\Lambda^{1},\quad*(\alpha\wedge\Phi)=J\alpha=-\alpha\lrcorner
F,\quad\alpha\in\Lambda^{1},\\\
*(\alpha\wedge\Psi^{+})=-\alpha\lrcorner\Psi^{-}\quad\alpha\in\Lambda^{1},\qquad*(\alpha\wedge\Psi^{-})=\alpha\lrcorner\Psi^{+}\quad\alpha\in\Lambda^{1},\\\
*(\beta\wedge{\Psi^{+}})=\beta\lrcorner{\Psi^{-}},\quad\beta\in\Lambda^{2},\qquad*(\beta\wedge{\Psi^{-}})=-\beta\lrcorner{\Psi^{+}},\quad\beta\in\Lambda^{2}\end{split}$
(3.26)

Extending the action of the almost complex structure on the exterior algebra,
$J\beta=(-1)^{p}\beta(J.,\dots,J.)$ for a p-form $\beta\in\Lambda^{p}$, we
recall the decomposition of the exterior algebra under the action of $SU(3)$
from [10] (see also [67]). Let $V=T_{p}M$ be the vector space at a point $p\in
M$ and $V^{*}$ be the dual vector space. Obviously $SU(3)$ acts irreducibly on
$V^{*}$ and $\Lambda^{5}V^{*}$, while $\Lambda^{2}V^{*}$ and
$\Lambda^{3}V^{*}$ decompose as follows:

$\begin{split}\Lambda^{2}V^{*}=\Lambda^{2}_{1}V^{*}\oplus\Lambda^{2}_{6}V^{*}\oplus\Lambda^{2}_{8}V^{*};\\\
\Lambda^{3}V^{*}=\Lambda^{3}_{Re}V^{*}\oplus\Lambda^{3}_{Im}\oplus\Lambda^{3}_{6}V^{*}\oplus\Lambda^{3}_{12}V^{*},\end{split}$

where

$\begin{split}\Lambda^{2}_{1}V^{*}=\mathbb{R}.F\quad\Lambda^{2}_{6}V^{*}=\\{\phi\in\Lambda^{2}V^{*}|J\phi=-\phi\\}\\\
\Lambda^{2}_{8}V^{*}=\\{\phi\in\Lambda^{2}V^{*}|J\phi=\phi,\phi\wedge
F^{2}=0\\}\\\
\Lambda^{3}_{Re}V^{*}=\mathbb{R}\Psi^{+};\qquad\Lambda^{3}_{Im}=\mathbb{R}\Psi^{-};\\\
\Lambda^{3}_{6}V^{*}=\\{\alpha\wedge F|\alpha\in\Lambda^{1}V^{*}\\};\\\
\Lambda^{3}_{12}V^{*}=\\{\gamma\in\Lambda^{3}V^{*}|\gamma\wedge
F=\gamma\wedge\Psi^{+}=\gamma\wedge\Psi^{-}=0\\}.\end{split}$

Denote by $S^{2}_{-}$ the space of symmetric 2-tensors which are of type
(2,0)+(0,2) with respect to the almost complex structure $J$, i.e.

$h(X,Y)=h(Y,X),\qquad h(JX,JY)=-h(X,Y).$ (3.27)

It is known (see [10]) that the map
$\gamma:S^{2}_{-}\leftrightarrow\Lambda^{3}_{12}$ defined by

$\gamma(h_{ij})=h_{ip}{\Psi^{+}}_{pjk}+h_{jp}{\Psi^{+}}_{pki}+h_{kp}{\Psi^{+}}_{pij},\qquad
h_{im}=\gamma^{-1}(B_{ijk})=\frac{1}{4}B_{ijk}{\Psi^{+}}_{mjk}$ (3.28)

is an isomorphism of $su(3)$ representations.

Writting

$F=\frac{1}{2}F_{ij}e_{ij},\quad\Psi^{+}=\frac{1}{6}\Psi^{+}_{ijk}e_{ijk},\quad\Psi^{-}=\frac{1}{6}\Psi^{-}_{ijk}e_{ijk},\quad\Phi=\frac{1}{24}\Phi_{ijkl}e_{ijkl}$

we have the obvious identites (c.f. [10])

$\begin{split}\Phi_{jslm}=F_{js}F_{lm}+F_{sl}F_{jm}+F_{lj}F_{sm},\\\
\Psi^{+}_{ipq}F_{pq}=0=\Psi^{-}_{ipq}F_{pq},\quad\Phi_{ijkl}\Psi^{+}_{jkl}=0=\Phi_{ijkl}\Psi^{-}_{jkl},\\\
F_{ip}F_{pj}=-\delta_{ij},\quad\Psi^{+}_{ijs}F_{sk}=-\Psi^{-}_{ijk},\quad\Psi^{-}_{ijs}F_{sk}=\Psi^{+}_{ijk},\\\
\Psi^{+}_{ipq}\Psi^{-}_{jpq}=-4F_{ij},\quad\Psi^{+}_{ipq}\Psi^{+}_{jpq}=4\delta_{ij}=\Psi^{-}_{ipq}\Psi^{-}_{jpq},\\\
\Psi^{+}_{kls}\Psi^{-}_{ijs}=\delta_{kj}F_{li}+\delta_{li}F_{kj}-\delta_{ki}F_{lj}-\delta_{lj}F_{ki},\\\
\Psi^{+}_{kls}\Psi^{+}_{ijs}=F_{kj}F_{li}-\delta_{li}\delta_{kj}-F_{ki}F_{lj}+\delta_{lj}\delta_{ki}\\\
\Phi_{ijkl}F_{kl}=4F_{ij},\quad\Phi_{ijkl}\Psi^{+}_{klp}=2\Psi^{+}_{ijp},\quad\Phi_{ijkl}\Psi^{-}_{klp}=2\Psi^{-}_{ijp},\\\
\Phi_{ijkl}\Psi^{+}_{lqp}=-F_{ij}\Psi^{-}_{pqk}-F_{jk}{\Psi^{-}}_{pqi}-F_{ki}{\Psi^{-}}_{pqj},\\\
\Phi_{ijkl}\Psi^{-}_{lqp}=F_{ij}\Psi^{+}_{pqk}+F_{jk}{\Psi^{+}}_{pqi}+F_{ki}{\Psi^{+}}_{pqj},\\\
\Phi_{ijkl}\Phi_{rjkl}=12\delta_{ir},\quad\Phi_{ijkl}\Phi_{klqr}=2F_{ij}F_{qr}-2\delta_{jq}\delta_{ir}+2\delta_{iq}\delta_{jr},\\\
\end{split}$ (3.29)

Now, we prove the following algebraic fact which is crucial for our
investigations.

######  Proposition 3.1.

Let $A$ be a 4-form and define the 3-form $B=(X\lrcorner A)$ for any $X\in
T_{p}M$. If the 3-form $B\in\Lambda^{3}_{12}$ then the four form $A$ vanishes
identically, $A=0$

###### Proof.

According to (3.28) and (3.27) the tensor

$C(X,Y,Z)=4\gamma^{-1}(X\lrcorner
A)(Y,Z)=A(X,Y,e_{i},e_{j}){\Psi^{+}}(Z,e_{i},e_{j})\\\
=C(X,Z,Y)=-C(Z,X,Y)=-C(Z,Y,X)=C(Y,Z,X)=C(Y,X,Z)=-C(X,Y,Z)$ (3.30)

and therefore vanishes. We get from (3.30) using (3.29)

$0=A_{pqkl}{\Psi^{+}}_{kls}{\Psi^{+}}_{ijs}=A_{pqkl}\Big{[}F_{kj}F_{li}-\delta_{li}\delta_{kj}-F_{ki}F_{lj}+\delta_{lj}\delta_{ki}\Big{]}=2A_{pqkl}F_{kj}F_{li}-2A_{pqji}$
(3.31)

The equalities (3.30) and (3.31) yield

$A(X,Y,JZ,JV)=A(X,Y,Z,V)=A(X,JY,JZ,V)=A(X,JY,Z,JV)=-A(X,Y,JZ,JV)$

Hence $A=0$ which completes the proof of Proposition 3.1. ∎

##  4 Almost Calabi-Yau with torsion space in dimension 6

The presence of a parallel spinor with respect to a metric connection with
torsion 3-form in dimension 6 leads to the reduction to $SU(3)$, ie the
existence of an almost hermitian structure and a linear connection preserving
the almost hermitian structure with torsion 3-form with holonomy group inside
SU(3).

The reduction of the holonomy group of the torsion connection to $SU(3)$, was
investigated intensively in the complex case, i.e. when the induced almost
complex structure is integrable ($N=0$) and it is known as the Strominger
condition. The Strominger condition (see [86]) has different very usefull
expressions in terms of the five torsion classes (see e.g. [13, 68, 30, 31,
54]). We present here the result from [54] which gives necessary and
sufficient condition to have a $\nabla$-parallel spinor in dimension 6,

######  Theorem 4.1.

[54, Theorem 4.1] Let $(M,g,J,\Psi)$ be a 6-dimensional smooth manifold with
an $SU(3)$-structure $(g,J,\Psi)$. The next two conditions are equivalent

  1. a)

$(M,g,J,\Psi)$ is an ACYT space, i.e. there exists a unique $SU(3)$-connection
${\nabla}$ with torsion 3-form whose holonomy is contained in SU(3),
${\nabla}g={\nabla}J={\nabla}\Psi^{+}={\nabla}\Psi^{-}=0$.

  2. b)

The Nijenhuis tensor $N$ is totally-skew symmetric and the following two
conditions hold

$\begin{split}d\Psi^{+}=\theta\wedge\Psi^{+}-\frac{1}{4}(N,\Psi^{+})*F;\qquad
d\Psi^{-}=\theta\wedge\Psi^{-}-\frac{1}{4}(N,\Psi^{-})*F.\end{split}$ (4.32)

The torsion is given by

$\begin{split}T=-*dF+*(\theta\wedge
F)+\frac{1}{4}(N,\Psi^{+})\Psi^{+}+\frac{1}{4}(N,\Psi^{-})\Psi^{-}.\end{split}$
(4.33)

Our first observation is the following

######  Proposition 4.2.

Let $(M,g,J,\Psi)$ be an ACYT 6-manifold.The following conditions are
equivalent:

  * a)

The exterior derivative of the Lee form is $J$-invariant,
$d\theta(X,Y)=d\theta(JX,JY);$

  * b)

The Nijenhuis tensor is of constant norm, $||N||^{2}=const.$;

  * c)

The Nijenhuis tensor is parallel with respect to the tosion connection,
${\nabla}N=0$

In particular, on a 6-dimensional CYT manifold the 2-form $d\theta$ is of type
(1,1).

###### Proof.

For the torsion $T$, we obtain from (4.33) using (3.26) that

$\begin{split}T=-*dF+*(\theta\wedge
F)+\frac{1}{4}(N,\Psi^{+})\Psi^{+}+\frac{1}{4}(N,\Psi^{-})\Psi^{-}.\\\
=*d*\Phi-\theta\lrcorner\Phi+\frac{1}{4}(N,\Psi^{+})\Psi^{+}+\frac{1}{4}(N,\Psi^{-})\Psi^{-}.\\\
=-\delta\Phi-\theta\lrcorner\Phi+\frac{1}{4}(N,\Psi^{+})\Psi^{+}+\frac{1}{4}(N,\Psi^{-})\Psi^{-}\end{split}$
(4.34)

The formula for the codifferential of a 4-form in terms of the Levi-Civita
connection ${\nabla}^{g}$ and (2.4) imply

$-\delta\Phi_{klm}={\nabla}^{g}_{j}\Phi_{jklm}={\nabla}_{j}\Phi_{jklm}-\frac{1}{2}T_{jsk}\Phi_{jslm}+\frac{1}{2}T_{jsl}\Phi_{jskm}-\frac{1}{2}T_{jsm}\Phi_{jskl}\\\
=-\frac{1}{2}T_{jsk}\Phi_{jslm}+\frac{1}{2}T_{jsl}\Phi_{jskm}-\frac{1}{2}T_{jsm}\Phi_{jskl}.$
(4.35)

since ${\nabla}\Phi=0$.

Substitute (4.35) into (4.34) to get the expression for the 3-form torsion
$T$,

$T_{klm}=-\frac{1}{2}T_{jsk}\Phi_{jslm}+\frac{1}{2}T_{jsl}\Phi_{jskm}-\frac{1}{2}T_{jsm}\Phi_{jskl}-\theta_{s}\Phi_{sklm}+\lambda{\Psi^{+}}_{klm}+\mu{\Psi^{-}}_{klm},$
(4.36)

where the functions $\lambda,\mu$ are defined by the scalar products

$\lambda=\frac{1}{4}(N,{\Psi^{+}})=\frac{1}{24}N_{ijk}{\Psi^{+}}_{ijk},\qquad\mu=\frac{1}{4}(N,{\Psi^{-}})=\frac{1}{24}N_{ijk}{\Psi^{-}}_{ijk}.$
(4.37)

It is straightforward to check that the Lee form $\theta$ can be expressed in
terms of the torsion $T$, the 2-form $F$ and the 4-form $\Phi$ as follows

$\theta_{i}=\frac{1}{2}T_{sjk}F_{jk}F_{si}=\frac{1}{6}T_{jkl}\Phi_{jkli};\quad\theta_{i}F_{iq}=-\frac{1}{2}T_{qjk}F_{jk}.$
(4.38)

Further, we calculate from (4.36) the functions $\lambda$ and $\mu$ in terms
if the torsion $T$ using (3.29) as follows

$\begin{split}T_{klm}{\Psi^{+}}_{klm}=-\frac{3}{2}T_{jsk}\Phi_{jslm}{\Psi^{+}}_{klm}+24\lambda=-3T_{jsk}{\Psi^{+}}_{jsk}+24\lambda\Longrightarrow\lambda=\frac{1}{6}T_{klm}{\Psi^{+}}_{klm};\\\
T_{klm}{\Psi^{-}}_{klm}=-\frac{3}{2}T_{jsk}\Phi_{jslm}{\Psi^{-}}_{klm}+24\mu=-3T_{jsk}{\Psi^{-}}_{jsk}+24\mu\Longrightarrow\mu=\frac{1}{6}T_{klm}{\Psi^{-}}_{klm}.\end{split}$
(4.39)

We have from (4.36) using the first formula in (3.29) and (4.38) that

$\begin{split}T_{klm}=-\frac{1}{2}T_{jsk}\Phi_{jslm}+\frac{1}{2}T_{jsl}\Phi_{jskm}-\frac{1}{2}T_{jsm}\Phi_{jskl}-\theta_{s}\Phi_{sklm}+\lambda{\Psi^{+}}_{klm}+\mu{\Psi^{-}}_{klm}\\\
=\theta_{s}F_{sk}F_{lm}+T_{jsk}F_{jm}F_{ls}-\theta_{s}F_{sl}F_{km}-T_{jsl}F_{jm}F_{ks}-\theta_{s}F_{sm}F_{lk}-T_{jsm}F_{jk}F_{ls}\\\
-\theta_{s}[F_{sk}F_{lm}+F_{kl}F_{sm}+F_{ls}F_{km}]+\lambda{\Psi^{+}}_{klm}+\mu{\Psi^{-}}_{klm}\\\
=T_{jsk}F_{jm}F_{ls}-T_{jsl}F_{jm}F_{ks}-T_{jsm}F_{jk}F_{ls}+\lambda{\Psi^{+}}_{klm}+\mu{\Psi^{-}}_{klm};\\\
\end{split}$ (4.40)

We can write the formula (4.40) in the form

$T(X,Y,Z)=T(JX,JY,Z)+T(JX,Y,JZ)+T(X,JY,JZ)+\lambda{\Psi^{+}}(X,Y,Z)+\mu{\Psi^{-}}(X,Y,Z).$
(4.41)

Hence, we get from (2.15) and (4.41)

$\begin{split}N=\lambda{\Psi^{+}}+\mu{\Psi^{-}},\quad||N||^{2}=24(\lambda^{2}+\mu^{2}).\end{split}$
(4.42)

######  Lemma 4.3.

On a 6-dimenionsl ACYT manifold the next identities hold

$\displaystyle d\mu=Jd\lambda,\quad d\lambda=-Jd\mu$ (4.43) $\displaystyle
d\theta(X,Y)-d\theta(JX,JY)=-(d\mu\lrcorner{\Psi^{-}})(X,Y)=-(d\lambda\lrcorner{\Psi^{+}})(X,Y).$
(4.44)

###### Proof.

Using (3.29), we obtain from (4.32)

$\begin{split}0=d^{2}{\Psi^{+}}=d\theta\wedge{\Psi^{+}}+\lambda\theta\wedge*F-d\lambda\wedge*F-\lambda
d*F=d\theta\wedge{\Psi^{+}}-\lambda\theta\wedge\Phi-\lambda*J\theta-d\lambda\wedge*F\\\
=d\theta\wedge{\Psi^{+}}-d\lambda\wedge*F,\\\
0=d^{2}{\Psi^{-}}=d\theta\wedge{\Psi^{-}}+\mu\theta\wedge*F-\mu
d*F-d\mu\wedge*F=d\theta\wedge{\Psi^{-}}-\mu\theta\wedge\Phi-\mu*J\theta-d\mu\wedge*F\\\
=d\theta\wedge{\Psi^{-}}-d\mu\wedge*F,\end{split}$ (4.45)

where we used the identity

$J\theta=\delta F=-*d*F=*(dF\wedge F),\quad*J\theta=-dF\wedge F=d*F.$

The equalities (4.45) and (3.26)imply

$d\theta_{ij}{\Psi^{-}}_{ijk}=2d\lambda_{s}F_{sk};\qquad
d\theta_{ij}{\Psi^{+}}_{ijk}=-2d\mu_{s}F_{sk}$ (4.46)

Multiply (4.46) with $F_{kl}$, use (3.29) and (4.46) to obtain

$-2d\mu_{s}F_{sl}=d\theta_{ij}{\Psi^{+}}_{ijl}=d\theta_{ij}{\Psi^{-}}_{ijk}F_{kl}=-2d\lambda_{l},$

which shows the identities (4.43) hold.

The equalities (4.46) yield applying (3.26) and (3.29)

$0=\frac{1}{2}d\theta_{ij}{\Psi^{+}}_{ijk}{\Psi^{+}}_{kab}+d\mu_{s}F_{sk}{\Psi^{+}}_{kab}=\frac{1}{2}d\theta_{ij}\Big{[}F_{ib}F_{ja}-F_{ia}F_{jb}-\delta_{ib}\delta_{ja}+\delta_{ia}\delta_{jb}\Big{]}+d\mu_{s}{\Psi^{-}}_{sab}\\\
=-d\theta_{ij}F_{ia}F_{jb}+d\theta_{ab}+d\mu_{s}{\Psi^{-}}_{sab}.$

This proves the equality (4.43). The equality (4.44) follows from already
established (4.43) and (3.26). The lemma is proved. ∎

The equivalence between a) and c) is a consequences from (4.44) and (4.42)
since ${\nabla}{\Psi^{+}}={\nabla}{\Psi^{-}}=0$.

Clearly, c) implies b). For the converse, the condition $||N||^{2}=const.$ and
(4.42) imply

$\lambda^{2}+\mu^{2}=const.,\lambda d\lambda+\mu d\mu=0$

which yields applying (4.43)

$-\lambda Jd\mu+\mu Jd\lambda=0,\quad\lambda d\mu-\mu d\lambda=0.$

Thus we obtain a homogeneous linear system with respect to $d\lambda$ and
$d\mu$. The determinant of this system is the constant $\lambda^{2}+\mu^{2}$.
If this constant iz zero then $N=0$ and (4.42) gives $\lambda=\mu=0$. If the
constant is not zero one concludes $d\lambda=d\mu=0$. Therefore ${\nabla}N=0$
due to (4.42). ∎

Applying the Hopf’s maximum principle we obtain

######  Theorem 4.4.

Let $(M,g,J,\Psi=\Psi^{+}+\sqrt{-1}\Psi^{-})$ be a 6-dimensional compact ACYT
manifold.

Then the Nijenhuis tensor is parallel with respect to the tosion connection
and exterior derivative of the Lee form is $J$-invariant,

${\nabla}N=0,\qquad d\theta(X,Y)=d\theta(JX,JY)$

In particular, the Nijenhuis tensor is of constant norm, $||N||^{2}=const.$

###### Proof.

From Lemma 4.3, we have $0=d^{2}\lambda=-dJd\mu$. The trace yields
$(dJd\mu)_{ij}F_{ij}=0$ which is an elliptic second order operator anihilating
constants and we apply the Hopf’s maximum principle (see e.g. [89, 35]) to
conclude $\mu=const$. Hence $d\lambda=d\mu=0$ due to (4.43). Now Proposition
4.2 completes the proof. ∎

######  Corollary 4.5.

Let $(M,g,J,\Psi)$ be a 6-dimensional ACYT manifold.

If the Lee form is closed, $d\theta=0$, then the Nijenhuis tensor is
${\nabla}$-parallel, ${\nabla}N=0$.

If the Lee form vanishes, $\theta=0$ then the Nijenhuis tensor $N$ is parallel
with respect to the torsion connection, the torsion 3-form is co-closed and
the Ricci tensor of ${\nabla}$ is symmetric,

${\nabla}N=\delta T=0,\qquad Ric(X,Y)=Ric(Y,X).$

###### Proof.

If $d\theta=0$, one has ${\nabla}N=0$ due to Poposition 4.2.

Suppose $\theta=0$. Then $d\lambda=d\mu=0$ due to (4.44) and (4.33) together
with (4.32) and (3.26) imply

$d*T=\lambda d{\Psi^{-}}-\mu d{\Psi^{+}}=(-\lambda\mu+\mu\lambda)\Phi=0.$

This compltes the proof in view of (2.13). ∎

As a consequence of the proof of Proposition 4.2 and Corollary 4.5 we
characterize Ricci-flat balanced ACYT.

######  Theorem 4.6.

Let $(M,g,J,\Psi)$ be a 6-dimensional balanced ACYT manifold, $\theta=0$.

The following conditions are equivalent:

  * a)

The Ricci tensor of the torsion connection vanishes, $Ric=0$.

  * b)

The torsion 3-form is harmonic, i.e. it is closed and co-closed, $dT=\delta
T=0$.

###### Proof.

Let the assumption a) holds. The condition $Ric=0$ implies $\delta T=0$.

On an $SU(3)$ manifold in dimension six, we can express the Ricci tensor in a
more appropriate form as follows. Since the torsion connection ${\nabla}$
preserves the $SU(3)$-structure its curvature lies in the Lie algebra $su(3)$,
i.e. it satisfies

$\begin{split}R_{ijab}{\Psi^{+}}_{abc}=R_{ijab}{\Psi^{-}}_{abc}=R_{ijab}F_{ab}=0\Longleftrightarrow
R_{ijab}\Phi_{abkl}=-2R_{ijkl}.\end{split}$ (4.47)

We have from (4.47) using (2.9), (4.38) and (2.5) that the Ricci tensor $Ric$
of ${\nabla}$ is given by

$Ric_{ij}=\frac{1}{2}R_{iabc}\Phi_{jabc}=\frac{1}{6}\Big{[}R_{iabc}+R_{ibca}+R_{icab}\Big{]}\Phi_{jabc}=\frac{1}{12}dT_{iabc}\Phi_{jabc}+\frac{1}{6}{\nabla}_{i}T_{abc}\Phi_{jabc}\\\
=\frac{1}{12}dT_{iabc}\Phi_{jabc}-{\nabla}_{i}\theta_{j}.$ (4.48)

Similarly, we have

$\begin{split}0=R_{iabc}{\Psi^{+}}_{abc}=\frac{1}{3}\Big{[}R_{iabc}+R_{ibca}+R_{icab}\Big{]}{\Psi^{+}}_{abc}=\frac{1}{6}dT_{iabc}{\Psi^{+}}_{abc}+\frac{1}{3}{\nabla}_{i}T_{abc}{\Psi^{+}}_{abc}\\\
0=R_{iabc}{\Psi^{-}}_{abc}=\frac{1}{3}\Big{[}R_{iabc}+R_{ibca}+R_{icab}\Big{]}{\Psi^{-}}_{abc}=\frac{1}{6}dT_{iabc}{\Psi^{-}}_{abc}+\frac{1}{3}{\nabla}_{i}T_{abc}{\Psi^{-}}_{abc}\end{split}$
(4.49)

From Proposition 4.5 we know that the condition $\theta=0$ imply
$d\lambda=d\mu=0$. Then $Ric=0$, (4.48) and (4.49) yield

$\begin{split}dT_{pjkl}\Phi_{jkli}=12{\nabla}_{p}\theta_{i}=0,\qquad
dT_{pjkl}{\Psi^{+}}_{jkl}=-12{\nabla}_{p}\lambda=0,\qquad
dT_{pjkl}{\Psi^{-}}_{jkl}=-12{\nabla}_{p}\mu=0.\end{split}$ (4.50)

The identities (4.50) show that the 3-form $(X\lrcorner
dT)\in\Lambda^{3}_{12}$. Then the 4-form $dT=0$ due to Proposition 3.1.

For the converse, if we assume b) then (4.48) and $\theta=0$ show $Ric=0$
which completes the proof. ∎

One gets from (4.48) and Theorem 2.4

######  Theorem 4.7.

Let $(M,g,J,\Psi)$ be a 6-dimensional ACYT manifold and the torsion connection
${\nabla}$ has curvature $R\in S^{2}\Lambda^{2}$ i.e. (2.11) holds.

Then the covariant derivative of the 1-form $J\theta$ with respect to
${\nabla}$ is skew-symmetric,

$({\nabla}_{X}J\theta)Y+({\nabla}_{Y}J\theta)X=0\Longleftrightarrow({\nabla}_{X}\theta)JY=-({\nabla}_{Y}\theta)JX.$

In particular, the vector field dual to the 1-form $J\theta$ is Killing vector
field.

  * a)

If in addition, the Nijenhuis tensor $N$ is of constant norm,
$||N||^{2}=const.$ then $N$ is ${\nabla}$-parallel, the covariant derivative
of the Lee form $\theta$ with respect to ${\nabla}$ is symmetric and
$J$-invariant,

${\nabla}N=0,\qquad({\nabla}_{X}\theta)Y=({\nabla}_{Y}\theta)X=({\nabla}_{JX}\theta)JY.$
(4.51)

  * b)

If the Lee form vanishes, $\theta=0$, i.e. the manifold is balanced $G_{1}$
space, $M\in W_{1}\oplus W_{3}$, then

${\nabla}T={\nabla}dF={\nabla}N=0,\qquad
Scal^{g}=-\frac{1}{3}||dF^{+}||^{2}+\frac{5}{64}||N||^{2}=const.$

###### Proof.

We get from (4.38) using the fact proved in [55, Lemma 3.4] that the curvature
of a metric connection ${\nabla}$ with skew-symmetric torsion $T$ satisfies
(2.11) exactly when ${\nabla}T$ is a 4-form,

$\begin{split}{\nabla}_{p}\theta_{i}F_{iq}=-\frac{1}{2}{\nabla}_{p}T_{qjk}F_{jk}=\frac{1}{2}{\nabla}_{q}T_{pjk}F_{jk}=-{\nabla}_{q}\theta_{i}F_{ip};\\\
({\nabla}_{X}\theta)JY=-({\nabla}_{Y}\theta)JX;\\\
({\nabla}_{X}J\theta)Y=-({\nabla}_{X}\theta)JY=({\nabla}_{Y}\theta)JX=-({\nabla}_{Y}J\theta)X.\end{split}$
(4.52)

It follows from (4.52) that the vector field $J\theta$ dual to the 1-form
$J\theta$ is Killing since (2.4) yields
$({\nabla^{g}}_{X}J\theta)Y+({\nabla^{g}}_{Y}J\theta)X=0$ which proves the
first part of Theorem 4.7.

Take the covariant derivative of the torsion expressed in (4.36) with respect
to ${\nabla}$ to get

${\nabla}_{p}T_{klm}=-\frac{1}{2}{\nabla}_{p}T_{jsk}\Phi_{jslm}+\frac{1}{2}{\nabla}_{p}T_{jsl}\Phi_{jskm}-\frac{1}{2}{\nabla}_{p}T_{jsm}\Phi_{jskl}\\\
-{\nabla}_{p}\theta_{s}\Phi_{sklm}+{\nabla}_{p}\lambda{\Psi^{+}}_{klm}+{\nabla}_{p}\mu{\Psi^{-}}_{klm},$
(4.53)

where we used that all the structure forms $g,F,{\Psi^{+}},{\Psi^{-}},\Phi$
are ${\nabla}$-parallel.

The fact that ${\nabla}T$ is a 4-form, taking the trace in (4.53) and using
(4.38) yields

$0={\nabla}_{p}T_{plm}=\frac{1}{2}{\nabla}_{p}T_{jsl}\Phi_{jspm}-\frac{1}{2}{\nabla}_{p}T_{jsm}\Phi_{jspl}-{\nabla}_{p}\theta_{s}\Phi_{splm}+{\nabla}_{p}\lambda{\Psi^{+}}_{plm}+{\nabla}_{p}\mu{\Psi^{-}}_{plm}\\\
=-3{\nabla}_{l}\theta_{m}+3{\nabla}_{m}\theta_{l}+{\nabla}_{p}\lambda{\Psi^{+}}_{plm}+{\nabla}_{p}\mu{\Psi^{-}}_{plm}+\frac{1}{2}({\nabla}_{p}\theta_{s}-{\nabla}_{s}\theta_{p})\Big{(}F_{ps}F_{lm}+F_{sl}F_{pm}+F_{lp}F_{sm}\Big{)}\\\
=-3{\nabla}_{l}\theta_{m}+3{\nabla}_{m}\theta_{l}+{\nabla}_{p}\lambda{\Psi^{+}}_{plm}+{\nabla}_{p}\mu{\Psi^{-}}_{plm}+\frac{1}{2}({\nabla}_{p}\theta_{s}-{\nabla}_{s}\theta_{p})\Big{(}F_{sl}F_{pm}+F_{lp}F_{sm}\Big{)}\\\
=-3{\nabla}_{l}\theta_{m}+3{\nabla}_{m}\theta_{l}+{\nabla}_{p}\lambda{\Psi^{+}}_{plm}+{\nabla}_{p}\mu{\Psi^{-}}_{plm}+({\nabla}_{p}\theta_{s}-{\nabla}_{s}\theta_{p})F_{sl}F_{pm},$
(4.54)

where we used the general identity ${\nabla}_{i}\theta_{s}F_{si}=0$ which is a
consequence of the fact that the 1-form $J\theta$ is co-closed. Indeed, we
have
$0=\delta(J\theta)=-{\nabla^{g}}_{i}(J\theta)_{i}=-{\nabla}_{i}(J\theta)_{i}=-{\nabla}_{i}\theta_{s}F_{si}$
applying (2.4).

Sunstitute (4.52) into (4.54) to conclude

$d^{{\nabla}}\theta_{lm}={\nabla}_{l}\theta_{m}-{\nabla}_{m}\theta_{l}=\frac{1}{2}[{\nabla}_{p}\lambda{\Psi^{+}}_{plm}+{\nabla}_{p}\mu{\Psi^{-}}_{plm}]={\nabla}_{p}\lambda{\Psi^{+}}_{plm}={\nabla}_{p}\mu{\Psi^{-}}_{plm},$
(4.55)

where we used (4.43) to conclude the last equality.

The condition $||N||^{2}=const.$ and Proposition 4.2 imply $d\lambda=d\mu=0$
and one gets $d^{{\nabla}}\theta=0$ from (4.55). Then (4.52) completes the
prove of a) in Theorem 4.7.

To show the condition $\theta=0$ yields ${\nabla}T=0$ we observe that (4.44)
implies the functions $\lambda$ and $\mu$ are constants. In particular
${\nabla}N=0$ due to (4.42). Then (4.39) together with (4.38) imply

$\begin{split}{\nabla}_{p}T_{jkl}\Phi_{jkli}=6{\nabla}_{p}\theta_{i}=0,\qquad{\nabla}_{p}T_{jkl}{\Psi^{+}}_{jkl}=6{\nabla}_{p}\lambda=0,\qquad{\nabla}_{p}T_{jkl}{\Psi^{-}}_{jkl}=6{\nabla}_{p}\mu=0.\end{split}$
(4.56)

The identities (4.56) show that the 3-form
$(X\lrcorner{\nabla}T)\in\Lambda^{3}_{12}$. Hence, the 4-form ${\nabla}T=0$
due to Proposition 3.1. Then the formulas (2.15) and (2.22) complete the proof
of the Theorem 4.7. ∎

###  4.1 Example

We take the next example from [54] which supports Theorem 4.7.

Let $G$ be the six-dimensional connected simply connected and nilpotent Lie
group, determined by the left invariant 1-forms $\\{e_{1},\dots,e_{6}\\}$ such
that

$\displaystyle de_{2}=de_{3}=de_{6}=0,$ (4.57) $\displaystyle
de_{1}=e_{3}\wedge e_{6},\quad de_{4}=e_{2}\wedge e_{6},\quad
de_{5}=e_{2}\wedge e_{3}.$

Consider the metric on $G\cong\mathbb{R}^{6}$ defined by
$g=\sum_{i=1}^{6}e_{i}^{2}$. Let $(F,\Psi)$ be the $SU(3)$-structure on $G$
given by (3.25). Then$(G,F,\Psi)$ is an almost complex manifold with a
$SU(3)$-structure.

It is easy to verify using (2.14) and (4.57) that

$\displaystyle dF=-3e_{236},\quad N=-\Psi^{-},\quad
d\Psi^{-}=*F,\quad(N,\Psi^{-})=-4,$ (4.58)
$\displaystyle\theta=d\Psi^{+}=(N,\Psi^{+})=0.$

Hence, $(G,\Psi,g,J)$ is neither complex nor Nearly Kähler manifold but it
fulfills the conditions (4.32) of Theorem 4.1 and therefore there exists a
$SU(3)$ connection with torsion 3-form on $(G,\Psi,g,J)$. The expression
(4.33) and (4.58) give

$T=-2e_{145}+e_{136}+e_{235}-e_{246},\quad
dT=-2(e_{1256}+e_{3456}+e_{1234})=2*F.$ (4.59)

The equalities (4.59) and (2.15) imply the nonzero essential terms of the
torsion connection are (c.f. [54])

$\displaystyle\nabla_{e_{1}}e_{6}=-e_{3},\qquad\nabla_{e_{5}}e_{2}=e_{3},\qquad\nabla_{e_{4}}e_{6}=-e_{2},$
(4.60)
$\displaystyle\nabla_{e_{4}}e_{5}=-e_{1},\qquad\nabla_{e_{5}}e_{1}=-e_{4},\qquad\nabla_{e_{1}}e_{4}=-e_{5}.$

It is easy to verify cf. [54]) that the curvature of the torsion connection
satisfies (2.11) and the torsion tensor $T$ as well as the Nijenhuis tensor
$N$ and $dF$ are parallel with respect to the connection $\nabla$.

The coefficient of the structure equations of the Lie algebra given by (4.57)
are integers. Therefore, the well-known theorem of Malcev [69] states that the
group $G$ has a uniform discrete subgroup $\Gamma$ such that
$Nil^{6}=G/\Gamma$ is a compact 6-dimensional nil-manifold. The
$SU(3)$-structure, described above, descends to $Nil^{6}$ and therefore we
obtain a compact example.

##  5 Proof of Theorem 1.3and Theorem 1.4

We begin with the next

######  Lemma 5.1.

Let $(M,g,J,\Psi)$ be an ACYT 6-manifold satisfying the condition (1.1).

Then the Lee form $\theta$ is ${\nabla}$-parallel, ${\nabla}\theta=0.$

In particular, the Lee form is co-slosed, $\delta\theta=0$.

###### Proof.

We already know from Theorem 4.7 a) that $d\lambda=d\mu=0$ and (4.51) holds.

Since ${\nabla}T$ is a 4-form, substitute (2.5) into (2.20) to get

$0=Ric(X,Y)+({\nabla}_{X}\theta)(Y)-({\nabla}_{X}T)(JY,e_{i},Je_{e})-\frac{1}{2}\sigma^{T}(X,JY,e_{i},Je_{i})\\\
=Ric(X,Y)+({\nabla}_{X}\theta)(Y)+2({\nabla}_{X}\theta)Y-\frac{1}{2}\sigma^{T}(X,JY,e_{i},Je_{i})\\\
=Ric(X,Y)+3({\nabla}_{X}\theta)(Y))-\frac{1}{2}\sigma^{T}(X,JY,e_{i},Je_{i}).$
(5.61)

Let $Ric=0$. Then we have from (2.20), (5.61) and (2.18)

$\begin{split}-\frac{1}{2}({\nabla}_{X}T)(JY,e_{i},Je_{i})=(\nabla_{X}\theta)Y=\frac{1}{4}dT(X,JY,e_{i},Je_{e})=\frac{1}{6}\sigma^{T}(X,JY,e_{i},Je_{i})\\\
=-\frac{1}{3}T(X,JY,J\theta)-\frac{1}{3}T(X,e_{i},e_{j})T(JY,Je_{i},e_{j}).\end{split}$
(5.62)

Using that
$d^{{\nabla}}\theta(X,Y)=({\nabla}_{X}\theta)Y-({\nabla}_{Y}\theta)X=-(d\mu\lrcorner{\Psi^{-}})(X,Y)=0$
due to (4.44) and (4.51), we get

$d\theta(X,Y)=({\nabla}^{g}_{X}\theta)Y-({\nabla}^{g}_{Y}\theta)X=-(d\mu\lrcorner{\Psi^{-}})(X,Y)+T(X,Y,\theta),$

which combined with (4.44) and (4.51) yields

$T(X,Y,\theta)-T(JX,JY,\theta)=-3d^{{\nabla}}\theta(X,Y)=-3(d\mu\lrcorner{\Psi^{-}})(X,Y)=-3(d\lambda\lrcorner{\Psi^{+}})(X,Y)=0.$
(5.63)

We have from (5.62), the symmetricity of ${\nabla}\theta$ and (5.63) that

$({\nabla}_{X}\theta)\theta=({\nabla}_{\theta}\theta)X=-\frac{1}{3}T(\theta,JX,J\theta)-\frac{1}{3}T(\theta,e_{i},e_{j})T(JX,Je_{i},e_{j})=-\frac{1}{3}T(\theta,e_{i},e_{j})T(JX,Je_{i},e_{j})\\\
=\frac{1}{3}T(\theta,Je_{i},e_{j})T(JX,e_{i},e_{j})=-\frac{1}{3}T(\theta,e_{i},Je_{j})T(JX,e_{i},e_{j})=-({\nabla}_{\theta}\theta)X=-({\nabla}_{X}\theta)\theta,$

which shows ${\nabla}||\theta||^{2}=2({\nabla}_{X}\theta)\theta=0$ and
$|\theta|^{2}=|J\theta|^{2}=const.$

Since $|J\theta|^{2}=const.$ we have

$\begin{split}0=\nabla_{e_{i}}\nabla_{e_{i}}||J\theta||^{2}=(\nabla_{e_{i}}\nabla_{e_{i}}J\theta)(e_{j}).J\theta(e_{j})+||\nabla
J\theta||^{2}.\end{split}$ (5.64)

We use the Killing condition of $J\theta$ and the Ricci identities for
${\nabla}$ to evaluate the first term in (5.64)

$(\nabla_{e_{i}}\nabla_{e_{i}}J\theta)(e_{j}).J\theta(e_{j})=-(\nabla_{e_{i}}\nabla_{e_{j}}J\theta)(e_{i}).J\theta(e_{j})\\\
=-(\nabla_{e_{j}}\nabla_{e_{i}}J\theta)(e_{i}).J\theta(e_{j})+R(e_{i},e_{j},e_{i},e_{k})J\theta(e_{k})J\theta(e_{j})+T(e_{i},e_{j},e_{k})(\nabla_{e_{k}}J\theta)e_{i}.J\theta(e_{j})\\\
=-Ric(J\theta,J\theta)+T(e_{i},J\theta,e_{k})(\nabla_{e_{k}}J\theta)e_{i}=T(e_{i},J\theta,e_{k})(\nabla_{e_{k}}J\theta)e_{i}=-T(e_{i},J\theta,e_{k})(\nabla_{e_{k}}\theta)Je_{i}\\\
=T(Je_{i},J\theta,e_{k})(\nabla_{e_{k}}\theta)e_{i}.$ (5.65)

We calculate from (5.63), (4.51) and (5.62)

$0=(\nabla_{e_{i}}T)(X,e_{i},\theta)+T(X,e_{i},e_{j})(\nabla_{e_{i}}\theta)(e_{j})=(\nabla_{e_{i}}T)(JX,Je_{i},\theta)+T(JX,Je_{i},e_{j})(\nabla_{e_{i}}\theta)(e_{j})\\\
=\frac{1}{3}\sigma^{T}(JX,Je_{i},\theta,e_{i})+T(JX,Je_{i},e_{j})(\nabla_{e_{i}}\theta)(e_{j})\\\
=\frac{1}{3}\Big{[}T(JX,Je_{i},e_{k})T(e_{k},\theta,e_{i})+T(Je_{i},\theta,e_{k})T(e_{k},JX,e_{i})+T(\theta,JX,e_{k})T(e_{k},Je_{i},e_{i})\Big{]}\\\
-T(Je_{i},JX,e_{j})(\nabla_{e_{j}}\theta)(e_{i})=\frac{2}{3}\Big{[}T(JX,Je_{i},e_{k})T(e_{k},\theta,e_{i})-T(\theta,JX,J\theta)\Big{]}-T(Je_{i},JX,e_{j})(\nabla_{e_{j}}\theta)(e_{i})\\\
=\frac{2}{3}T(JX,Je_{i},e_{k})T(e_{k},\theta,e_{i})-T(Je_{i},JX,e_{j})(\nabla_{e_{j}}\theta)(e_{i}).$
(5.66)

Set $X=\theta$ into (5.66) to get applying (5.63)

$3T(Je_{i},J\theta,e_{j})(\nabla_{e_{j}}\theta)(e_{i})=2T(J\theta,Je_{i},e_{k})T(\theta,e_{i},e_{k})=-2T(J\theta,e_{i},e_{k})T(\theta,Je_{i},e_{k})\\\
=2T(J\theta,e_{i},e_{k})T(\theta,e_{i},Je_{k})=-2T(J\theta,e_{i},Je_{k})T(\theta,e_{i},e_{k})\\\
=-3T(Je_{i},J\theta,e_{j})(\nabla_{e_{j}}\theta)(e_{i}).$

Hence,

$T(Je_{i},J\theta,e_{j})(\nabla_{e_{j}}\theta)(e_{i})=0.$ (5.67)

Substitute (5.67) into (5.64) to get ${\nabla}J\theta=0={\nabla}\theta$ which
completes the proof of Lemma 5.1 ∎

To finish the proof of Theorem 1.3 we observe from (5.62), (4.38), (4.39),
(4.48), (4.49) and Lemma 5.1 the validity of the following identities

$\begin{split}2{\nabla}_{p}T_{jkl}\Phi_{jkli}=12{\nabla}_{p}\theta_{i}=0=dT_{pjkl}\Phi_{jkli};\\\
2{\nabla}_{p}T_{jkl}{\Psi^{+}}_{jkl}=12{\nabla}_{p}\lambda=0=-dT_{pjkl}{\Psi^{+}}_{jkl};\\\
2{\nabla}_{p}T_{jkl}{\Psi^{-}}_{jkl}=12{\nabla}_{p}\mu=0=-dT_{pjkl}{\Psi^{-}}_{jkl}.\end{split}$
(5.68)

The identities (5.68) show that the 3-forms
$(X\lrcorner{\nabla}T)\in\Lambda^{3}_{12}$ and $(X\lrcorner
dT)\in\Lambda^{3}_{12}$. Hence, the four forms ${\nabla}T=dT=0$ due to
Proposition 3.1 and (2.5) implies $\sigma^{T}=0$. Therefore, ${\nabla^{g}}T=0$
because of (2.6). Now Theorem 2.4 shows that the Riemannian first Bianchi
identity (2.10) holds.

For the converse, it is well known algebraic fact that the Riemannian first
Bianchi identity (2.10) on an ACYT manifold implies (2.11) and the vanishing
of the Ricci tensor. This completes the proof of Theorem 1.3. Theorem 1.4
follows from Theorem 1.3 and Theorem 4.4.

##  6 ACYT with closed torsion and generalized Ricci solitons

######  Theorem 6.1.

Let $(M,g,J,\Psi)$ be an ACYT 6-manifold. The following conditions are
equivalent:

  * a).

The torsion is closed, $dT=0$.

  * b).

The exterior derivative of the Lee form belongs to $su(3)\cong\Lambda^{2}_{8}$
(i.e. it is $J$-invariant trace-free 2-orm) and the Ricci tensor of the
torsion connection is equal to $-{\nabla}\theta$,

$d\theta(X,Y)=d\theta(JX,JY),\quad d\theta\lrcorner
F=d\theta\wedge\Phi=0,\qquad Ric=-{\nabla}\theta$ (6.69)

  * c).

The Nijenhuis tensor is parallel with respect to the tosion connection and the
Ricci tensor of the torsion connection is equal to $-{\nabla}\theta$,

${\nabla}N=0,\qquad\qquad Ric=-{\nabla}\theta$ (6.70)

###### Proof.

The condition $dT=0$ and the equality (4.48) imply $Ric=-{\nabla}\theta$. We
get from (4.49) and (4.39) $d\lambda=d\mu=0$ since the torsion is closed.
Hence, c) follows from (4.42).

Now Proposition 4.2 implies the first and the third identities in (6.69).

To show that the second equality in (6.69) follows from c) we observe $-\delta
T_{ij}=Ric_{ij}-Ric_{ji}=d^{{\nabla}}\theta_{ij}$. Consequently, we have
$d^{{\nabla}}\theta_{ij}F_{ij}={\nabla}_{i}\theta_{j}F_{ij}=0$ since $\delta
J\theta={\nabla}_{i}\theta_{j}F_{ij}=\delta^{2}F=0$. Hence,

$0=(d^{{\nabla}}\theta\lrcorner F).vol=-(\delta T\lrcorner F).vol=\delta
T\wedge\Phi=\frac{1}{2}\delta T\wedge F\wedge F.$ (6.71)

On the other hand, we have from (4.33) using $d\lambda=d\mu=0$ that

$d*T=d\theta\wedge F-\theta\wedge dF+\lambda d{\Psi^{-}}-\mu d{\Psi^{+}}\\\
=d\theta\wedge F-\theta\wedge
dF+\lambda\Big{[}\theta\wedge{\Psi^{-}}-\mu*F\Big{]}-\mu\Big{[}\theta\wedge{\Psi^{+}}-\lambda*F\Big{]}\\\
=d\theta\wedge F-\theta\wedge
dF+\lambda\theta\wedge{\Psi^{-}}-\mu\theta\wedge{\Psi^{+}},$

yielding the identity

$\begin{split}d*T\wedge F=d\theta\wedge F\wedge F-\theta\wedge dF\wedge
F=d\theta\wedge F\wedge F+\theta\wedge(*J\theta)=d\theta\wedge F\wedge F;\\\
d\theta\wedge F\wedge F=d*T\wedge F=-d*T\wedge*\Phi=\Phi\wedge\delta
T=0,\end{split}$

where we apply (6.71) to obtain the last equality. Hence, b) follows from c).

For the converse, assume b) holds. The third equality in (6.69) combined with
(4.48) yields

$dT_{jabc}\Phi_{iabc}=0.$

The first equality in (6.69) implies $d\lambda=d\mu=0$ due to Proposition 4.2.
Apply (4.49) and (4.39) to get

$dT_{jabc}{\Psi^{+}}_{abc}=dT_{jabc}{\Psi^{-}}_{abc}=0.$

The last three equalities show that the 3-form $(X\lrcorner
dT)\in\Lambda^{3}_{12}$. Hence, the four form $dT=0$ due to Proposition 3.1
which completes the equivalences between a) and b). ∎

In view of Theorem 4.4 we get from Theorem 6.1

######  Corollary 6.2.

Let $(M,g,J,\Psi)$ be a compact ACYT 6-manifold.The following conditions are
equivalent:

  * a).

The torsion is closed, $dT=0$.

  * b).

The Ricci tensor of the torsion connection satisfies $Ric=-{\nabla}\theta$.

In particular $d\theta\in su(3)\cong\Lambda^{2}_{8}$.

As a consequence of Theorem 6.1, we obtain

######  Corollary 6.3.

Let $(M,g,J,\Psi)$ be a CYT 6-manifold. he following conditions are
equivalent:

  * a).

The CYT space is pluriclosed, $dT=\partial\bar{\partial}F=0$.

  * b).

The Ricci tensor of the torsion connection satgisfies $Ric=-{\nabla}\theta$.

In particular $d\theta\in su(3)\cong\Lambda^{2}_{8}$.

###  6.1 Proof of Theorem 1.1

###### Proof.

The equivances of a) and c) as well as between d) and f) follow from Theorem
6.1.

The second Bianchi identity for the torsion connection reads [60, Proposition
3.5]

$d(Scal)_{j}-2{\nabla}_{i}Ric_{ji}+\frac{1}{6}d||T||^{2}_{i}+\delta
T_{ab}T_{abj}+\frac{1}{6}T_{abs}dT_{jabc}=0.$ (6.72)

From Theorem 6.1 we have taking into account (2.13)

$Ric=-{\nabla}\theta,\quad Scal=\delta\theta,\quad\delta
T_{ij}={\nabla}_{i}\theta_{j}-{\nabla}_{j}\theta_{i}.$ (6.73)

Substitute (6.73) into (6.72) and use $dT=0$ to get

${\nabla}_{j}\delta\theta+2{\nabla}_{i}{\nabla}_{j}\theta_{i}+\delta
T_{ab}T_{abj}+\frac{1}{6}d||T||^{2}_{j}=0.$ (6.74)

We evaluate the second term in (6.74) using the Ricci identity for ${\nabla}$
and (6.73) as follows

${\nabla}_{i}{\nabla}_{j}\theta_{i}={\nabla}_{j}{\nabla}_{i}\theta_{i}-R_{ijis}\theta_{s}-T_{ijs}{\nabla}_{s}\theta_{i}=-{\nabla}_{j}\delta\theta-{\nabla}_{j}\theta_{s}.\theta_{s}-\frac{1}{2}\delta
T_{si}T_{sij}.$ (6.75)

We obtain from (6.75) and(6.74)

$-{\nabla}_{j}\delta\theta-2{\nabla}_{j}\theta_{s}.\theta_{s}+\frac{1}{6}{\nabla}_{j}||T||^{2}=0.$
(6.76)

Another covariant derivative of (6.76) together with (6.73) yield

$\Delta\delta\theta-2{\nabla}_{j}{\nabla}_{j}\theta_{s}.\theta_{s}-2||Ric||^{2}-\frac{1}{6}\Delta||T||^{2}=0,$
(6.77)

where $\Delta$ is the Laplac operator
$\Delta=-{\nabla^{g}}_{i}{\nabla^{g}}_{i}f=-{\nabla}_{i}{\nabla}_{i}f$ acting
on smooth function $f$ since the torsion of ${\nabla}$ is a 3-form.

On the other hand, (6.72) yields

${\nabla}_{i}{\nabla}_{j}\theta_{i}={\nabla}_{i}({\nabla}_{i}\theta_{j}-\delta
T_{ij})={\nabla}_{i}{\nabla}_{i}\theta_{j}-{\nabla}_{i}\delta
T_{ij}={\nabla}_{i}{\nabla}_{i}\theta_{j}-\frac{1}{2}\delta T_{ia}T_{iaj},$
(6.78)

where we used the identity

${\nabla}_{i}\delta T_{ij}=\frac{1}{2}\delta T_{ia}T_{iaj}.$ (6.79)

shown in [60, Proposition 3.2] for any metric connection with skew torsion.

We give a proof of (6.79) for completness. The identity $\delta^{2}=0$
together with (2.4) and (2.6) imply

$0=\delta^{2}T_{k}={\nabla^{g}}_{i}{\nabla^{g}}_{j}T_{ijk}={\nabla^{g}}_{i}{\nabla}_{j}T_{ijk}={\nabla}_{i}\delta
T_{ik}+\frac{1}{2}T_{iks}\delta T_{is}={\nabla}_{i}\delta
T_{ik}-\frac{1}{2}\delta T_{is}T_{isk}.$

The equalities (6.78) and (6.74) yield

${\nabla}_{j}\delta\theta+2{\nabla}_{i}{\nabla}_{i}\theta_{j}+\frac{1}{6}{\nabla}_{j}||T||^{2}=0.$
(6.80)

Substitute the second term in (6.80) into (6.77) to get

$\Delta\Big{(}\delta\theta-\frac{1}{6}||T||^{2}\Big{)}+\theta_{j}{\nabla}_{j}\Big{(}\delta\theta+\frac{1}{6}||T|\Big{)}|^{2}=||Ric||^{2}\geq
0.$ (6.81)

Assume b) holds. Since $M$ is compact and $d||T||^{2}=0$ we may apply the
strong maximum principle (see e.g. [89, 35]) to achieve
$\delta\theta=const.=0$. Conversely, the condition $\delta\theta=0$ imply
$d||T||^{2}=0$ by the strong maximum principle applied to (6.81). Hence b) is
equivalent to f).

Further, assume f) or b). Then (2.21) implies $d||\theta||^{2}=0$ since
$dT=\delta\theta=0=d||T||^{2}$ and the norm of the Nijenhuis tensor is
constant. We calculate using the Ricci identity for ${\nabla}$, (6.73) and
(6.79)

$\begin{split}0=\frac{1}{2}{\nabla}_{i}{\nabla}_{i}||\theta||^{2}=\theta_{j}{\nabla}_{i}{\nabla}_{i}\theta_{j}+||{\nabla}\theta||^{2}=\theta_{j}{\nabla}_{i}\Big{(}{\nabla}_{j}\theta_{i}+\delta
T_{ij}\Big{)}+||{\nabla}\theta||^{2}\\\ =-\theta_{j}{\nabla}_{j}\delta\theta-
R_{ijis}\theta_{s}\theta_{j}-\theta_{j}T_{ijs}{\nabla}_{s}\theta_{i}+\theta_{j}{\nabla}_{i}\delta
T_{ij}+||{\nabla}\theta||^{2}\\\
=-{\nabla}_{j}\theta_{s}\theta_{s}\theta_{j}-\frac{1}{2}\theta_{j}\delta
T_{si}T_{sij}+\frac{1}{2}\theta_{j}\delta
T_{ia}T_{iaj}+||{\nabla}\theta||^{2}\\\
=-\frac{1}{2}\theta_{j}{\nabla}_{j}||\theta||^{2}+||{\nabla}\theta||^{2}=||{\nabla}\theta||^{2}\end{split}$

Hence, we show that c) follows from b).

For the converse, suppose c) holds, ${\nabla}\theta=0$. Then b) is a
consequence from (6.76) and (6.73).

Finally, to show the equivalences of b) and e) we use (2.13) and (6.73) to
write (6.81) in the form

$\Delta\Big{(}Scal^{g}-\frac{5}{12}||T||^{2}\Big{)}+\theta_{j}{\nabla}_{j}\Big{(}Scal^{g}-\frac{1}{12}||T||^{2}|\Big{)}=||Ric||^{2}\geq
0.$ (6.82)

The maximum principle applied to (6.82) imply b) is equivalent to e) in the
same way as above.

The proof of Theorem 1.1 is completed.

∎

###  6.2 Generalized steady Ricci solitons

It is shown in [35, Proposition 4.28] that a Riemannian manifold $(M,g,T)$
with a closed torsion $dT=0$ is a steady generalized Ricci soliton if there
exists a vector field $X$ and a two form $B$ such that it is a solution to the
equations

$Ric^{g}=\frac{1}{4}T^{2}-\mathbb{L}_{X}g,\qquad\delta T=B,$ (6.83)

for $B$ satisfying $d(B+X\lrcorner T)=0.$ In particular
$\Delta_{d}T=\mathbb{L}_{X}T$.

We also recall the equivalent formulation [35, Definition 4.31] that a compact
Riemannian manifold $(M,g,T)$ with a closed 3-form $T$ is a steady generalized
Ricci soliton with $k=0$ if there exists a vector field $X$ such that

$Ric^{g}=\frac{1}{4}T^{2}-\mathbb{L}_{X}g,\qquad\delta T=-X\lrcorner T,\qquad
dT=0.$ (6.84)

If the vector field $X$ is a gradient of a smooth function $f$ then one has
the notion of a generalized gradient Ricci soliton.

It is shown in [35, Proposition 8.14] that any compact pluriclosed (strong)
CYT space is automatically a steady generalized Ricci soliton.

We extend this result to 6-dimensional ACYT space with a closed torsion form.

######  Proposition 6.4.

Let $(M,g,J,\Psi)$ be a 6-dimensional ACYT manifold with closed torsion form,
$dT=0$.

Then it is a steady generalized Ricci soliton with $X=\theta$ and
$B=d\theta-\theta\lrcorner T$.

###### Proof.

As we identify the vector field $X$ with its corresponding 1-form via the
metric, we have using (2.4)

$dX_{ij}={\nabla^{g}}_{i}X_{j}-{\nabla^{g}}_{j}X_{i}={\nabla}_{i}X_{j}-{\nabla}_{j}X_{i}+X_{s}T_{sij}$
(6.85)

In view of (2.13), (2.4) and (6.85) we write the first equation in (6.83) in
the form

$Ric_{ij}=-\frac{1}{2}\delta
T_{ij}-\frac{1}{2}({\nabla}_{i}X_{j}+{\nabla}_{j}X_{i})=-\frac{1}{2}\delta
T_{ij}-{\nabla}_{i}X_{j}+\frac{1}{2}dX_{ij}-\frac{1}{2}X_{s}T_{sij}$ (6.86)

Set $X=\theta,\quad B=d\theta-\theta\lrcorner T$ and use (6.69) to get $\delta
T=B$ and (6.86) is trivially satisfied. Hence, (6.83) holds since
$d(B+\theta\lrcorner T)=d^{2}\theta=0$. ∎

One fundamental consequence of Perelman’s energy formula for Ricci flow is
that compact steady solitons for Ricci flow are automatically gradient.
Adapting these energy functionals to generalized Ricci flow it is proved in
[35, Chapter 6] that steady generalized Ricci solitons on compact manifolds
are automatically gradient with k = 0, i.e there exists a smooth function $f$
such that $X=grad(f)$ and (6.84) takes the form

$Ric^{g}_{ij}=\frac{1}{4}T^{2}_{ij}-{\nabla^{g}}_{i}{\nabla^{g}}_{j}f,\qquad\delta
T_{ij}=-df_{s}T_{sij},\qquad dT=0$ (6.87)

and the smooth function $f$ is determined with $u=\exp(-\frac{1}{2}f)$ where
$u$ is the first eigenfunction of the Schrödinger operator

$-4\Delta+Scal^{g}-\frac{1}{12}||T||^{2}=-4\Delta+Scal+\frac{1}{6}||T||^{2},$
(6.88)

where the Laplac operator is defind by $\Delta=-\delta d$ (see [35, Lemma 6.3,
Corollary 6.10, Corollary 6.11]).

In terms of the torsion connection (6.87) can be written in the form (see
[60])

$Ric_{ij}=-{\nabla}_{i}{\nabla}_{j}f,\qquad\delta T_{ij}=-df_{s}T_{sij},\qquad
dT=0.$ (6.89)

Thus, a pluriclosed CYT space is a steady generalized gradient Ricci soliton
[33, Proposition 2.6], (see also [35, Proposition 8.14]).

We show that the similar conclusions hold for a compact closed ACYT manifold
of dimension six.

######  Theorem 6.5.

Let $(M,g,J,\Psi)$ be a 6-dimensional compact ACYT manifold with closed
torsion, $dT=0$. The next two conditions are equivalent.

  * a).

$(M,g,J,\Psi)$ is a steady generalized gradient Ricci soliton. i.e. there
exists a smooth function $f$ determined with $u=\exp(-\frac{1}{2}f)$ where $u$
is the first eigenfunction of the Schrödinger operator (6.88) such that
(6.87), equivalently (6.89), hold.

  * b).

For $f$ determined by the first eigenfunction $u$ of the Schrödinger operator
(6.88) with $u=\exp(-\frac{1}{2}f)$, the vector field

$V=\theta-df\quad is\quad{\nabla}-parallel,\quad{\nabla}V=0.$

The ${\nabla}$-parallel vector field $V$ determines $d\theta$ and preserves
the $SU(3)$ structure $(g,J,F,{\Psi^{+}},{\Psi^{-}})$ and the Nijenhuis
tensor, i.e

$d\theta=V\lrcorner
T,\quad\mathbb{L}_{V}g=\mathbb{L}_{V}F=\mathbb{L}_{V}J=\mathbb{L}_{V}{\Psi^{+}}=\mathbb{L}_{V}{\Psi^{-}}=\mathbb{L}_{V}N=0.$
(6.90)

The vector field $JV$ is also ${\nabla}$ parallel and therefore Killing.

###### Proof.

The first statement a) follows from Proposition 6.4 and the general
considerations in [35, Chapter 6], [35, Lemma 6.3, Corollary 6.10, Corollary
6.11]).

To prove b) follows from a) observe that the condition $dT=0$ and (2.13) imply

$Ric=-{\nabla}\theta,\quad-\delta
T=-d^{{\nabla}}\theta=-d\theta+\theta\lrcorner T$

by Theorem 6.1 and (2.13). The latter combined with (6.89) yield

${\nabla}(\theta-df)=0,\qquad d\theta=V\lrcorner T.$

For the converse, b) yields ${\nabla}\theta={\nabla}df$, which, combined with
Theorem 6.1 and (2.13) gives

$Ric_{ij}=-{\nabla}_{i}\theta_{j}=-{\nabla}_{i}{\nabla}_{j}f,\quad-\delta
T_{ij}=Ric_{ij}-Ric_{ji}=df_{s}T_{sij}.$

since
$0=d^{2}f={\nabla^{g}}_{i}{\nabla^{g}}_{j}f-{\nabla^{g}}_{j}{\nabla^{g}}_{i}f={\nabla}_{i}{\nabla}_{j}f-{\nabla}_{j}{\nabla}_{i}f+df_{s}T_{sij}$.
Hence, (6.89) holds which proves the equivalences between a) and b).

Further, ${\nabla}JV=({\nabla}V)J=0$. We also have
$(\mathbb{L}_{V}g)_{ij}={\nabla^{g}}_{i}V_{j}+{\nabla^{g}}_{j}V_{i}={\nabla}_{i}V_{j}+{\nabla}_{j}V_{i}=0$
since ${\nabla}V=0$. Hence $V$ and $JV$ are Killing vector fields.

We calculate from the definitions of the Lie derivative and the torsion tensor
(see e.g. [66]) using ${\nabla}V=0$ that

$(\mathbb{L}_{V}F)(X,Y)=VF(X,Y)-F([V,X],Y)-F(X,[V,Y])\\\
=VF(X,Y)-F({\nabla}_{V}X,Y)-F(X,{\nabla}_{V}Y)+T(V,X,e_{a})F(e_{a},Y)+T(V,Y,e_{a})F(X,e_{a})\\\
=({\nabla}_{V}F)(X,Y)+d\theta(X,JY)-d\theta(Y,JX)=0,$ (6.91)

where we apply the already proved identity $d\theta=V\lrcorner T$ to get the
third equality and use ${\nabla}F=0$ and that $d\theta$ is (1,1) form due to
Theorem 6.1 to achieve the last identity.

Using the general identity

$g(\mathbb{L}_{V}J)X,Y)=-(\mathbb{L}_{V}g)(JX,Y)-(\mathbb{L}_{V}F)(X,Y)$

we conclude $\mathbb{L}_{V}J=0$.

Similarly to the proof of (6.91) we obtain

$\begin{split}(\mathbb{L}_{V}{\Psi^{+}})_{ijk}=d\theta_{is}{\Psi^{+}}_{sjk}+d\theta_{js}{\Psi^{+}}_{ski}+d\theta_{ks}{\Psi^{+}}_{sij};\\\
(\mathbb{L}_{V}{\Psi^{-}})_{ijk}=d\theta_{is}{\Psi^{-}}_{sjk}+d\theta_{js}{\Psi^{-}}_{ski}+d\theta_{ks}{\Psi^{-}}_{sij}\end{split}$
(6.92)

We calculate from (6.92) using the identites (3.29) that

$\begin{split}(\mathbb{L}_{V}{\Psi^{+}})_{ijk}{\Psi^{+}}_{ijk}=3d\theta_{is}{\Psi^{+}}_{sjk}{\Psi^{+}}_{ijk}=12d\theta_{is}\delta_{si}=0=(\mathbb{L}_{V}{\Psi^{-}})_{ijk}{\Psi^{-}}_{ijk};\\\
(\mathbb{L}_{V}{\Psi^{+}})_{ijk}{\Psi^{-}}_{ijk}=3d\theta_{is}{\Psi^{+}}_{sjk}{\Psi^{-}}_{ijk}=-12d\theta_{is}F_{si}=0=-(\mathbb{L}_{V}{\Psi^{-}})_{ijk}{\Psi^{+}}_{ijk}\end{split}$
(6.93)

since $d\theta\in su(3)$ due to Theorem 6.1 which is used to conclude the last
equality in (6.93).

We know that $d\theta$ is a (1,1) two form due to Theorem 6.1 and
${\Psi^{+}},{\Psi^{-}}$ are (3,0)+(0,3) three forms. Apply this to (6.92) to
conclude that the three forms
$(\mathbb{L}_{V}{\Psi^{+}}),(\mathbb{L}_{V}{\Psi^{-}})$ are (3,0)+(0,3) three
forms and therefore belong to $span\\{{\Psi^{+}},{\Psi^{-}}\\}$ since M is six
dimensional. Therefore, (6.93) imply
$\mathbb{L}_{V}{\Psi^{+}}=0=\mathbb{L}_{V}{\Psi^{-}}$. Since $d\lambda=d\mu=0$
we get from (4.42) $\mathbb{L}_{V}N=0$ which completes the proof of (6.90). ∎

In the complex case $N=0$, one has general identity
$\mathbb{L}_{JV}J=-J\mathbb{L}_{V}J$ and Theorem 6.5 gives

######  Corollary 6.6.

Let $(M,g,J,\Psi)$ be a 6-dimensional compact pluriclosed CYT space. The next
two conditions are equivalent.

  * a).

$(M,g,J,\Psi)$ is a steady generalized gradient Ricci soliton.

  * b).

The vector field $V=\theta-df$ is ${\nabla}$-parallel, ${\nabla}V=0.$

The ${\nabla}$-parallel vector field $V$ determines $d\theta$ and preserves
the $SU(3)$ structure $(g,J,F,{\Psi^{+}},{\Psi^{-}})$, i.e

$d\theta=V\lrcorner
T,\quad\mathbb{L}_{V}g=\mathbb{L}_{V}F=\mathbb{L}_{V}J=\mathbb{L}_{V}{\Psi^{+}}=\mathbb{L}_{V}{\Psi^{-}}=0.$

The vector field $JV$ is ${\nabla}$ parallel, Killing and holomorphic.

This result slightly improves [33, Proposition 2.6] in dimension six.

Acknowledgements We would like to thank Jeffrey Streets and Ilka Agricola for
extremly useful remarks, comments and suggestions.

The research of S.I. is partially supported by Contract 80-10-192 / 17.5.2023
with the Sofia University ”St.Kl.Ohridski” and the National Science Fund of
Bulgaria, National Scientific Program “VIHREN”, Project KP-06-DV-7. The
research of N.S. is partially financed by the European Union-Next Generation
EU, through the National Recovery and Resilience Plan of the Republic of
Bulgaria, project SUMMIT BG-RRP-2.004-0008-C01.

## References

  * [1] I. Agricola, A. Ferreira, _Einstein manifolds with skew torsion_ , The Quarterly Journal of Mathematics, 65, Issue 3, (2014), 717-741, https://doi.org/10.1093/qmath/hat050
  * [2] I. Agricola, A. Ferreira, Th. Friedrich, _The classification of naturally reductive homogeneous spaces in dimensions $n\leq 6$_, Diff. Geom. Appl., 39 (2015), 59-92.
  * [3] I. Agricola and T. Friedrich, _A note on flat metric connections with antisymmetric torsion_ , Diff. Geom. Appl. 28 (2010), 480-487.
  * [4] B. Alexandrov, T. Friedrich and N. Schoemann, Almost Hermitian 6-manifolds revisited, J. Geom. Phys. 53 (2005) 1.
  * [5] B.Alexandrov, S.Ivanov, _Vanishing theorems on hermitian manifolds_ , Diff. Geom. Appl. 14 (2001), 251-265.
  * [6] D. Angella, A. Otal, L. Ugarte, R. Villacampa, _On Gauduchon connections with Kähler-like curvature_ , Comm. Anal. Geom. 30 (2022), no. 5, 961-1006.
  * [7] K. Becker, M. Becker, K. Dasgupta, P.S. Green, Compactifications of Heterotic Theory on Non-Kahler Complex Manifolds: I, JHEP 0304 (2003) 007.
  * [8] K. Becker, M. Becker, K. Dasgupta, P.S. Green, E. Sharpe, _C_ ompactifications of Heterotic Strings on Non-Kahler Complex Manifolds: II, Nucl. Phys. B678 (2004), 19-100.
  * [9] F. Belgun, A. Moroianu, _Nearly Kdhler manifolds with reduced holonomy_ , Ann. Global Anal. Geom., 19 (2001), pp. 307-319.
  * [10] Lucio Bedulli, Luigi Vezzoni, _The Ricci tensor of SU(3)-manifolds_ , J. Geom. Phys. 57 (2007), n. 4, 1125-1146.
  * [11] J. M. Bismut, _A local index theorem of non-Kaehler manifolds_ , Math. Ann. 284 (1989), 681-699.
  * [12] E. Cartan, J.A. Schouten, _On Riemannian manifolds admitting an absolute parallelism_ , Proc. Amsterdam 29 (1926), 933-946.
  * [13] G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis, G. Zoupanos, Non-Kaehler String Backgrounds and their Five Torsion Classes, Nucl.Phys. B652 (2003) 5-34.
  * [14] G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, BPS Action and Superpotential for Heterotic String Compactifications with Fluxes, JHEP 0310 (2003) 004.
  * [15] S. Chiossi, S. Salamon, The intrinsic torsion of SU(3) and $G_{2}$-structures, Differential Geometry, Valencia 2001, World Sci. Publishing, 2002, pp. 115-133.
  * [16] I. Chrysikos, _Invariant Connections with Skew-Torsion and ${\nabla}$-Einstein Manifolds_, Journal of Lie Theory 26 (2016) 011-048.
  * [17] I. Chrysikos, Ch. O’Cadiz Gustad, H. Winther, _Invariant connections and ${\nabla}$-Einstein structures on isotropy irreducible spaces_, J. Geom. Phys. 138 (2019) 257-284
  * [18] Richard Cleyton, Andrei Moroianu, Uwe Semmelmann, _Metric connections with parallel skew-symmetric torsion_ , Advances in Mathematics, Volume 378, 12 February 2021, 107519, https://doi.org/10.1016/j.aim.2020.107519
  * [19] Tristan C. Collins, Sebastien Picard, Shing-Tung Yau, _The Strominger system in the square of a Kaehler class_ , arXiv:211.03784.
  * [20] Tristan C. Collins, Sebastien Picard, Shing-Tung Yau, _Stability of the tangent bundle through conifold transitions_ , arXiv:2102.11170.
  * [21] K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G-flux, JHEP 081999023;
  * [22] Teng Fei, Zhijie Huang, Sebastien Picard, _The Anomaly flow over Riemann surfaces_ , International Mathematics Research Notices, Volume 2021, Issue 3, February 2021, Pages 2134-2165, https://doi.org/10.1093/imrn/rnz076
  * [23] Teng Fei, Zhijie Huang, Sebastien Picard, _A Construction of Infinitely Many Solutions to the Strominger System_ , J. Differential Geom. Volume 117, Number 1 (2021), 23-39.
  * [24] M. Ferna’ndez, S. Ivanov, L. Ugarte, R. Villacampa, _Non-Kaehler Heterotic String Compactifications with non-zero fluxes and constant dilaton_ , Comm. Math. Phys. 288 (2009), 677-697.
  * [25] A Fino, M Parton, S Salamon, _Families of strong KT structures in six dimensions_ , Comment. Math. Helv. 79 (2004), 317-340;
  * [26] A. Fino, A. Tomassini, _Blow-ups and resolutions of strong Kähler with torsion metrics_ , Adv. Math. 221 (2009), 914-935.
  * [27] A.Fino, L. Ugarte, On generalized gauduchon metrics, Proc. Edinb. Math. Soc. 56(3) 2013, 733-753.
  * [28] Th. Friedrich, S. Ivanov, _Parallel spinors and connections with skew-symmetric torsion in string theory_ , Asian Journ. Math. 6 (2002), 303 - 336.
  * [29] Th. Friedrich, S.Ivanov, Killing spinor equations in dimension 7 and geometry of integrable $G_{2}$ manifolds, J. Geom. Phys 48 (2003), 1-11.
  * [30] J. X. Fu and S. T. Yau, _The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampére equation_ , J. Differential Geom. 78 (2008), no. 3, 369-428.
  * [31] J. X. Fu and S. T. Yau, _The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation_ , J. Diff. Geom. 78 (2009), 369–428.
  * [32] J. Fu, Z. Wang, and D. Wu, Semilinear equations, the $\gamma_{k}$ function, and generalized Gauduchon metrics, J. Eur. Math. Soc., 15 (2013), 659-680.
  * [33] M. Garcia-Fernandez, J. Jordan, J. Streets, _Non-Kaehler Calabi-Yau geometry and pluriclosed flow_ , arXiv:2106.13716.
  * [34] M. Garcia-Fernandez, R. Rubio, C. S. Shahbazi, C. Tipler. _Canonical metrics on holomorphic Courant algebroids_ , Proc. London Math. Soc. 125 (2022), 700-758.
  * [35] M. Garcia-Fernandez and J. Streets, Generalized Ricci Flow. AMS University Lecture Series 76, 2021.
  * [36] S. J. Gates, Jr., C. M. Hull and M. Rocek, _Twisted Multiplets and New Supersymmetric Nonlinear Sigma Models_ , Nucl. Phys. B 248 (1984) 157.
  * [37] P. Gauduchon, _Hermitian connnections and Dirac operators_ , Boll. Un. Mat. It. 11-B (1997) Suppl. Fasc., 257-288.
  * [38] J. Gillard, G. Papadopoulos, D. Tsimpis, Anomaly, Fluxes and (2,0) Heterotic-String Compactifications, JHEP 0306 (2003) 035.
  * [39] E. Goldstein, S. Prokushkin, Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure, Commun. Math. Phys. 251 (2004), 65-78.
  * [40] A. Gray, _The structure of nearly Kaehler manifolds_ , Math. Ann., 223 (1976), pp. 233-248.
  * [41] A. Gray, L. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35–58.
  * [42] S. Gurrieri, J. Louis, A. Micu, D. Waldram, Mirror Symmetry in Generalized Calabi-Yau Compactifications, Nucl.Phys. B654 (2003) 61-113
  * [43] S. Gurrieri, A. Micu, Type IIB Theory on Half-flat Manifolds, Class.Quant.Grav. 20 (2003), 2181-2192.
  * [44] J.Gauntlett, D.Martelli, D.Waldram, Superstrings with Intrinsic torsion, Phys. Rev. D69 (2004) 086002.
  * [45] J.Gauntlett, N. Kim, D.Martelli, D.Waldram, Fivebranes wrapped on SLAG three-cycles and related geometry, JHEP 0111 (2001) 018.
  * [46] J.P. Gauntlett, D. Martelli, S. Pakis, D. Waldram, G-Structures and Wrapped NS5-Branes, Commun. Math. Phys. 247 (2004), 421-445.
  * [47] J. Gutowski and G. Papadopoulos, _Heterotic Black Horizons_ , JHEP 1007 (2010) 011
  * [48] J. Gutowski and G. Papadopoulos, _Heterotic horizons, Monge-Ampere equation and del Pezzo surfaces_ , JHEP 1010 (2010) 084
  * [49] J. Gutowski, S. Ivanov, G. Papadopoulos, Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class, Asian J. Math. 7 (2003), 39-80.
  * [50] A.S. Haupt, O. Lechtenfeld, E.T. Musaev, _Order $\ alpha^{\prime}$ heterotic domain walls with warped nearly Kähler geometry_, Journal of High Energy Physics, (JHEP), Volume 2014, article id. $\\#152$, 28 pp.
  * [51] P. S. Howe and G. Papadopoulos, _Twistor spaces for HKT manifolds_ , Phys. Lett. B 379 (1996) 80
  * [52] P. S. Howe and G. Sierra, _Two-dimensional Supersymmetric Nonlinear Sigma Models With Torsion_ , Phys. Lett. B 148 (1984) 451.
  * [53] C.M. Hull, _Compactification of the heterotic superstrings_ , Phys. Lett. B 178 (1986) 357-364
  * [54] P. Ivanov and S. Ivanov, _SU(3) instantons and G2, Spin(7) heterotic string solitons_ , Commun. Math. Phys. 259 (2005) 79-102.
  * [55] S. Ivanov, _Geometry of Quaternionic Kähler connections with torsion_ , J. Geom. Phys. 41 (2002), 235-257.
  * [56] S. Ivanov, Connection with torsion, parallel spinors and geometry of Spin(7) manifolds, Math. Res. Lett. 11 (2004), no. 2-3, 171–186.
  * [57] S. Ivanov and G. Papadopoulos, A no go theorem for string warped compactifications, Phys. Lett. B497  (2001) 309;
  * [58] S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Class. Quant. Grav. 18  (2001) 1089;
  * [59] S. Ivanov and G. Papadopoulos, Vanishing theorems on $(l|k)$-strong Kaehler manifolds with torsion, Adv. Math., _237_ (2013), 147-164.
  * [60] S. Ivanov, N. Stanchev, _The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons_ , arXiv:2307.03986.
  * [61] J. Jost and S.T. Yau, A non-linear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math 170 (1993) 221; Corrigendum Acta Math 177 (1994) 307.
  * [62] S. Kachru, M.B. Schulz, P.K. Tripathy, S.P. Trivedi, New Supersymmetric String Compactifications, JHEP 0303 (2003) 061
  * [63] V. Kirichenko, _K-spaces of maximal rank_ , (russian), Mat. Zam., 22 (1977), pp. 465-476.
  * [64] M. Klaput, A. Lukas, C,l Matti, _Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory_ , JHEP 1109 (2011) 100 .
  * [65] M. Klaput, A. Lukas, C. Matti and E.E. Svanes, _Moduli Stabilising in Heterotic Nearly Kaehler Compactifications_ , JHEP 01 (2013) 015
  * [66] Kobayashi, S.; Nomizu, K., Foundations of differential geometry. Vol. I, Wiley Classics Lib. Wiley-Intersci. Publ. John Wiley & Sons, Inc., New York, 1996, xii+329 pp.
  * [67] Magdalena Larfors, Andre Lukas, Fabian Ruehle, _Calabi-Yau Manifolds and SU(3) Structure_ , Journal of High Energy Physics 2019(1), DOI: 10.1007/JHEP01(2019)171
  * [68] J. Li, S.-T. Yau, The existence of supersymmetric string theory with torsion, J. Diff. Geom. 70 (2005) 143.
  * [69] A. I. Malcev, On a class of homogeneous spaces, reprinted in Amer. Math. Soc. Translations, Series 1, 9 (1962), 276-307.
  * [70] Lei Ni, Fangyang Zheng, _A classification of locally Chern homogeneous hermitian manifolds_ , arXiv:2301.00579.
  * [71] Xenia de la Ossa, Eirik E. Svanes, _Holomorphic Bundles and the Moduli Space of N=1 Heterotic Compactifications_ , J. High Energy Phys. 2014, no. 10, 123, front matter+54 pp.
  * [72] Xenia de la Ossa, Magdalena Larfors, Eirik E. Svanes, _Exploring SU(3) Structure Moduli Spaces with Integrable G2 Structures_ , Adv. Theor. Math. Physics, Volume 19 (2015) Number 4, 837-903.
  * [73] G. Papadopoulos (2,0)-supersymmetric sigma models and almost complex structures, Nucl.Phys. B448 (1995), 199-219.
  * [74] Duong H. Phong, Sebastien Picard, Xiangwen Zhang, _Anomaly flows_ , Comm. Anal. Geom. 26 (2018), no. 4, 955-1008.
  * [75] Duong H. Phong, Sebastien Picard, Xiangwen Zhang, _The anomaly flow and the Fu-Yau equation_ , Ann. PDE 4 (2018), no. 2, Paper No. 13, 60 pp.
  * [76] Duong H. Phong, Sebastien Picard, Xiangwen Zhang, _Geometric flows and Strominger systems_ , Math. Z. 288 (2018), no. 1-2, 101-113.
  * [77] Phong, D.H., Picard, S., Zhang, X., _The Fu-Yau equation with negative slope parameter_ , Invent. Math. 209 2 (2017), doi:10.1007/s00222-016-0715-z
  * [78] Duong H. Phong, Sebastien Picard, Xiangwen Zhang, _New curvature flows in complex geometry_ , Surveys in Differential Geometry 2017\. Celebrating the 50th anniversary of the Journal of Differential Geometry, 331-364, Surv. Differ. Geom., 22, Int. Press, Somerville, MA, 2018.
  * [79] Duong H. Phong, _Geometric flows from unified string theories_ , Contribution to Surveys in Differential Geometry, ”Forty Years of Ricci flow”, edited by H.D. Cao, R. Hamilton, and S.T. Yau, arXiv:2304.02533.
  * [80] D. Popovici, _Limits of projective manifolds under holomorphic deformations: Hodge numbers and strongly Gauduchon metrics_ , Invent. Math. 194 (2013), no. 3, 515-534.
  * [81] N. Schoemann, Almost hermitian structures with parallel torsion, J. Geom. Phys. 57 (2007) 2187.
  * [82] J. Streets, _Classification of solitons for pluriclosed flow on complex surfaces_ , Math. Ann. 375 (2019), no. 3-4, 1555-1595.
  * [83] J. Streets, G. Tian, _A parabolic flow of pluriclosed metrics_ , Int. Math. Res. Not. IMRN 16 (2010), 3101-3133.
  * [84] J. Streets, G. Tian, _Regularity results for pluriclosed flow_ , Geom. Topol. 17 (2013), no. 4, 2389-2429.
  * [85] J. Streets, G. Tian,, _Generalized Kaehler geometry and the pluriclosed flow_ , Nuclear Phys. B 858 (2012), no. 2, 366-376.
  * [86] A. Strominger, _Superstrings With Torsion_ , Nucl. Phys. B 274, 253 (1986).
  * [87] Q. Wang, B. Yang, F. Zheng, _On Bismut flat manifolds_ , Trans. Amer. Math. Soc. 373 (2020), no. 8, 5747-5772.
  * [88] K. Yano. Differential geometry on complex and almost complex spaces. International Series of Monographs in Pure and Applied Mathematics, Vol 49, A Pergamon Press Book. 1965.
  * [89] K. Yano and S. Bochner, Curvature and Betti Numbers, Ann. of Math. Studies 32, Princeton University Press, 1953.
  * [90] S. T. Yau, Q. Zhao, F. Zheng, _On Strominger Kähler-like manifolds with degenerate torsion_ , Trans. Amer. Math. Soc. 376 (2023), no. 5, 3063-3085.
  * [91] Yanan Ye, _Bismut-Einstein metrics on compact complex surfaces_ , arXiv:2212.04060.
  * [92] Q. Zhao, F. Zheng, _Strominger connection and pluriclosed metrics_ , J. Reine Angew. Math. 796 (2023), 245-267.
  * [93] Q. Zhao, F. Zheng, _Complex nilmanifolds and Kähler-like connections_ , J. Geom. Phys. 146 (2019), 103512, 9 pp.
  * [94] Q. Zhao, F. Zheng, _On Hermitian manifolds with Bismut-Srominger parallel torsion_ , ArXiv:2208.03071.
  * [95] Q. Zhao, F. Zheng, _Bismut Kaehler-like manifolds of dimension 4 and 5_ , arXiv:2303.09267.