#  On the existence of a generalized class of recurrent manifolds

Absos Ali Shaikh, Indranil Roy and Haradhan Kundu
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Dedicated to Prof. Dr.
Constantin Udrişte on his 76th birthday

###### Abstract.

The present paper deals with the proper existence of a generalized class of
recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have
established the proper existence of various generalized notions of recurrent
manifolds. For this purpose we have presented a metric and computed its
curvature properties and finally we have obtained the existence of a new class
of semi-Riemannian manifolds which are non-recurrent but hyper-generalized
recurrent, Ricci recurrent, conharmonically recurrent, manifold of recurrent
curvature 2-forms and semisymmetric; not weakly symmetric but weakly Ricci
symmetric, conformally weakly symmetric and conharmonically weakly symmetric;
non-Einstein but Ricci simple; do not satisfy $P\cdot P=0$ but fulfills the
condition $P\cdot P=-\frac{1}{3}Q(S,P)$, $P$ being the projective curvature
tensor.

###### Key words and phrases:

Locally symmetric, semisymmetric, recurrent, hyper-generalized recurrent,
pseudosymmetric, Ricci pseudosymmetric, curvature 2-forms

###### 2010 Mathematics Subject Classification:

53C15, 53C25, 53C35

##  1\. Introduction

Let $M$ (dim $M=n\geq 3$) be a connected semi-Riemannian manifold endowed with
a semi-Riemannian metric $g$ of signature ($s,n-s$), $0\leq s\leq n$. If $s=0$
or $n$ then $M$ is a Riemannian manifold, and if $s=1$ or $n-1$ then $M$ is a
Lorentzian manifold. A semi-Riemannian manifold has mainly three notions of
curvature, namely, the Riemann-Christoffel curvature tensor $R$ (simply called
curvature tensor), the Ricci tensor $S$ and the scalar curvature $r$. Let
$\nabla$ be the Levi-Civita connection on $M$, which is the unique torsion
free metric connection on $M$.
It is well known that $M$ is locally symmetric if its local geodesic
symmetries at each point are all isometry. In terms of curvature restriction,
$M$ is locally symmetric if $\nabla R=0$ ([5], [6]). The study of
generalization of locally symmetric manifolds was started in 1946 and
continued till date in different directions, such as semisymmetric manifolds
by Cartan [7] (which were latter classified by Szab$\acute{\mbox{o}}$ ([82,
83, 84])), $\kappa$-spaces or recurrent spaces by Ruse ([53], [54], [55]),
weakly symmetric manifolds by Selberg [56], pseudosymmetric manifolds by Chaki
[8], pseudosymmetric manifolds by Deszcz ([18], [24]), weakly symmetric
manifolds by Tam$\acute{\mbox{a}}$ssy and Binh [85] etc.
Again in the literature of differential geometry we find various curvature
restricted geometric structures formed by imposing a restriction on the Ricci
tensor, such as Ricci symmetric manifold ($\nabla S=0$), Ricci recurrent
manifold by Patterson [52], Ricci semisymmetric manifold by
Szab$\acute{\mbox{o}}$ ([82, 83, 84]), Ricci pseudosymmetric manifold by
Deszcz [16], pseudo Ricci symmetric manifold by Chaki [9], weakly Ricci
symmetric manifold by Tam$\acute{\mbox{a}}$ssy and Binh [86] etc.
During the study of simply harmonic Riemannian spaces, Ruse ([53], [54], [55])
introduced the notion of kappa spaces, which were latter known as recurrent
spaces (see, [55], [87]). The manifold $M$ is said to be recurrent ([53],
[54], [55]) (resp., Ricci recurrent [52]) if

$\nabla R=A\otimes R\ \ \ \left(\mbox{resp.}\ \nabla S=A\otimes S\ \right)$

holds for some 1-form $A$. During the last six decades, in literature we find
various generalized notions of recurrent manifolds, such as 2-recurrent
manifolds by Lichnerowicz [39], conformally recurrent manifolds by Adati and
Miyazawa [1], projectively recurrent manifolds by Adati and Miyazawa [2],
concircular recurrent or generalized recurrent manifolds by Dubey [12], quasi-
generalized recurrent manifolds by Shaikh and Roy [77], hyper-generalized
recurrent manifolds by Shaikh and Patra [76], weakly generalized recurrent
manifolds by Shaikh and Roy [78], super generalized recurrent manifolds by
Shaikh et al. [79], manifolds of recurrent curvature 2-forms ([4], [40]) etc.
Recently, Olszak and Olszak [51] showed that every generalized recurrent
manifold is recurrent (see also [48], [49], [50]). It is interesting to note
that although the notion of generalized recurrent manifold does not exist but
the proper existence of weakly generalized recurrent (resp., quasi-generalized
recurrent and super generalized recurrent) manifold is presented in [59]
(resp., [79]).
The study of recurrence condition is a study of tensors which are invariant
(upto a scaling) under a parallel transport. It is an important topic of
research as the recurrent condition implies that the curvature tensor remains
invariant under the covariant differentiation [55]. Recent investigation of
Hall and Kirik [35] shows that the study of recurrent manifolds is closely
related to the holonomy theory.
As a generalization of recurrent manifolds, recently Shaikh and Patra [76]
introduced the notion of hyper-generalized recurrent manifolds and studied its
various geometric properties. The present paper is devoted to the study of the
proper existence of hyper-generalized recurrent manifolds and the
interrelation of such notion with other geometric structures. The paper is
organized as follows. Section 2 deals with various curvature restricted
geometric structures and their interrelations as preliminaries. The last
section is devoted to the proper existence of hyper-generalized recurrent
manifold. For this purpose we pose the following question:
Q. 1 Does there exist a hyper-generalized recurrent manifold which is
(i) not locally symmetric but semisymmetric,
(ii) not weakly symmetric but weakly Ricci symmetric as well as conformally
weakly symmetric,
(iii) not recurrent but Ricci recurrent, conformally recurrent as well as a
manifold of recurrent curvature 2-forms,
(iv) neither class $\mathcal{A}$ nor $\mathcal{B}$ but of constant scalar
curvature,
(v) not Einstein but Ricci simple i.e., special quasi-Einstein,
(vi) does not satisfy $P\cdot P=0$ but fulfills the condition $P\cdot
P=LQ(S,P)$ for some scalar $L$.
In the last section of this paper we have computed the curvature properties of
the metric given by

$ds^{2}=g_{ij}dx^{i}dx^{j}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}+e^{x^{1}}(dx^{4})^{2},\
\ i,j=1,2,3,4,$

which produces the answer of Q. 1 as affirmative, and finally we obtain the
existence of a new class of semi-Riemannian manifolds (see Theorem 3.1 and
Remark 3.2).

##  2\. Preliminaries

Let us consider a connected semi-Riemannian manifold $M$ of dimension $n(\geq
3)$ (these conditions are assumed throughout the paper). The manifold $M$ is
said to be Einstein (briefly, $E_{n}$) if its Ricci tensor $S$ of type $(0,2)$
satisfies $S=\frac{r}{n}g$. The manifold is said to be quasi-Einstein
(briefly, $QE_{n}$) ([15], [19], [20], [28], [29], [30], [33], [67], [73],
[81]) if

$S-\alpha g=\beta\eta\otimes\eta$

on $\\{x\in M:\left(S-\frac{r}{n}\,g\right)_{x}\neq 0\\}$, where $\otimes$ is
the tensor product, $\eta$ is an 1-form and $\alpha,\beta$ are two scalars on
that set. It may be noted that if $\alpha$ vanishes identically then a quasi-
Einstein manifold turns into a Ricci simple manifold. Hence a semi-Riemannian
manifold $(M,g)$, $n\geq 3$, is said to be Ricci simple if the following
condition

(2.1) $S=\beta\eta\otimes\eta$

holds for a scalar $\beta$ and a 1-form $\eta$.

######  Proposition 2.1.

A manifold satisfying (2.1) is Ricci recurrent if $\eta$ is parallel (i.e.,
$\nabla\eta=0$).

######  Proposition 2.2.

A manifold satisfying $R=\alpha~{}g\wedge(\eta\otimes\eta)$ for a scalar
$\alpha$ and a 1-from $\eta$, then the manifold is recurrent if $\eta$ is
parallel. The result is also true for any other (0,4)-tensor.

The manifold $M$ is said to be of class $\mathcal{B}$ or of Codazzi type Ricci
tensor ([32], [58]) (resp. class $\mathcal{A}$ or cyclic Ricci parallel [34])
if

$(\nabla_{X_{1}}S)(X_{2},X_{3})=(\nabla_{X_{2}}S)(X_{1},X_{3})$
$\left(\mbox{resp.}\ \ \
(\nabla_{X_{1}}S)(X_{2},X_{3})+(\nabla_{X_{2}}S)(X_{3},X_{1})+(\nabla_{X_{3}}S)(X_{1},X_{2})=0\
\right)$

holds for all vector fields $X_{1},X_{2},X_{3}\in\chi(M)$, where $\chi(M)$
being the Lie algebra of all smooth vector fields on $M$. Throughout this
paper we consider $X,Y,X_{i}\in\chi(M)$, $i=1,2,3,\cdots$.
It is well known that the conformal transformation is an angle preserving
mapping, the projective transformation is a geodesic preserving mapping
whereas concircular transformation is the geodesic circle preserving mapping
and conharmonic transformation is the harmonic function preserving mapping and
each transformation induces a curvature tensor as invariant, namely, conformal
curvature tensor $C$, projective curvature tensor $P$, concircular curvature
tensor $W$ and conharmonic curvature tensor $K$ ([15], [21], [33], [37],
[88]), and are respectively given as:

$\displaystyle C(X_{1},X_{2},X_{3},X_{4})=\left[R-\frac{1}{n-2}(g\wedge
S)+\frac{r}{2(n-2)(n-1)}(g\wedge g)\right](X_{1},X_{2},X_{3},X_{4}),$
$\displaystyle P(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
R(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle-\frac{1}{n-1}[g(X_{1},X_{4})S(X_{2},X_{3})-g(X_{2},X_{4})S(X_{1},X_{3})],$
$\displaystyle W(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle\left[R-\frac{r}{2n(n-1)}(g\wedge
g)\right](X_{1},X_{2},X_{3},X_{4}),$ $\displaystyle
K(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle\left[R-\frac{1}{n-2}(g\wedge
S)\right](X_{1},X_{2},X_{3},X_{4}),$

where ‘$\wedge$’ denotes the the Kulkarni-Nomizu product. For two
$(0,2)$-tensors $J$ and $F$ the Kulkarni-Nomizu product is defined as (see
e.g. [21], [33], [38]):

$\displaystyle(J\wedge F)(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle J(X_{1},X_{4})F(X_{2},X_{3})+J(X_{2},X_{3})F(X_{1},X_{4})$
$\displaystyle-$ $\displaystyle
J(X_{1},X_{3})F(X_{2},X_{4})-J(X_{2},X_{4})F(X_{1},X_{3}).$

We recall that the manifold $M$ is locally symmetric (resp., Ricci symmetric)
([5], [6], [7]) if $\nabla R=0$ (resp., $\nabla S=0$). Similarly for other
curvature tensors we have conformally symmetric, projectively symmetric,
concircularly symmetric, conharmonically symmetric manifolds. We note that
local symmetry, projective symmetry and concircular symmetry are equivalent
(see [75] and also references therein). An $n$-dimensional locally symmetric
(resp., Ricci symmetric and conformally symmetric) manifold is denoted by
$S_{n}$ (resp., $RS_{n}$ and $CS_{n}$).
Again the manifold $M$ is said to be recurrent ([53], [54], [55], [87])
(resp., Ricci recurrent [52]) if

$\nabla R=A\otimes R\ \ \ \left(\mbox{resp.}\ \nabla S=A\otimes S\ \right)$

holds for a 1-form $A$ on $\\{x\in M:R_{x}\neq 0\\}$ (resp., $\\{x\in
M:S_{x}\neq 0\\}$). Similarly we can define the conformally recurrent,
projectively recurrent, concircularly recurrent, conharmonically recurrent
manifolds. Recently Shaikh and Kundu [75] showed that the recurrent,
projectively recurrent and concircularly recurrent manifolds are equivalent.
An $n$-dimensional recurrent (resp., Ricci recurrent and conformally
recurrent) manifold is denoted by $K_{n}$ (resp., $RK_{n}$ and $CK_{n}$).
The manifold $M$ is said to be quasi-generalized recurrent [77] (briefly,
$QGK_{n}$), hyper generalized recurrent [76] (briefly, $HGK_{n}$), weakly
generalized recurrent [78] (briefly, $WGK_{n}$) and super generalized
recurrent [79] (briefly, $SGK_{n}$) respectively if

$\nabla R=A\otimes R+B\otimes g\wedge[g+(\eta\otimes\eta)],$ $\nabla
R=A\otimes R+B\otimes(S\wedge g),$ $\nabla R=A\otimes
R+B\otimes\frac{1}{2}(S\wedge S)\mbox{ and}$ $\nabla R=A\otimes
R+B\otimes(S\wedge S)+D\otimes(g\wedge S)+E\otimes(S\wedge S)$

holds on $\\{x\in M:R_{x}\neq 0\mbox{ and
}[g\wedge(g+\eta\otimes\eta)]_{x}\neq 0\\}$, $\\{x\in M:R_{x}\neq 0\mbox{ and
}(g\wedge S)_{x}\neq 0\\}$, $\\{x\in M:R_{x}\neq 0\mbox{ and }(S\wedge
S)_{x}\neq 0\\}$ and $\\{x\in M:R_{x}\neq 0\mbox{ and either }(S\wedge
S)_{x}\neq 0\mbox{ or }(g\wedge S)_{x}\neq 0\\}$ respectively for some 1-forms
$A,B,D,E$ and $\eta$.
We note that on a quasi-Einstein manifold with associated scalars $\alpha$ and
$\beta$, if $\alpha=\beta$ then the notion of $WGK_{n}$ and $QGK_{n}$ are
equivalent, and if $2\alpha=\beta$ then $HGK_{n}$ and $QGK_{n}$ are
equivalent.
Again the manifold $M$ is said to be generalized Ricci recurrent if $\nabla
S=A\otimes S+B\otimes g$ holds on $\\{x\in M:S_{x}\neq 0\\}$ for some 1-forms
$A$ and $B$.
During the study of a certain kind of conformally flat Riemannian space of
class one, Sen and Chaki [57] obtained a curvature condition which was also
studied by Chaki [8] and named it as pseudosymmetric manifold. The manifold
$M$ is said to be pseudosymmetric in sense of Chaki or simply Chaki
pseudosymmetric [8] (briefly, $PS_{n}$) if

$\displaystyle(\nabla_{X}R)(X_{1},X_{2},X_{3},X_{4})=2A(X)R(X_{1},X_{2},X_{3},X_{4})+A(X_{1})R(X_{1},X,X_{3},X_{4})$
$\displaystyle\hskip
56.9055pt+A(X_{2})R(X_{1},X,X_{3},X_{4})+A(X_{3})R(X_{1},X_{2},X,X_{4})+A(X_{4})R(X_{1},X_{2},X_{3},X)$

holds $\forall~{}X,X_{i}\in\chi(M)$ and a non-zero 1-form $A$ on $\\{x\in
M:R_{x}\neq 0\\}$. Similarly we can define the conformally (briefly,
$CPS_{n}$) (resp., projectively, concircularly, conharmonically)
pseudosymmetric manifolds. We note that every recurrent manifold is Chaki
pseudosymmetric [31] but the result is not true for other curvature tensors
(e.g., see Theorem 3.1 for the existence of $CK_{n}$ which is not $CPS_{n}$).
Again in 1988, Chaki [9] introduced the notion of pseudo Ricci symmetric
manifolds. The manifold $M$ is said to be pseudo Ricci symmetric [9] (briefly,
$PRS_{n}$) if

$(\nabla_{X}S)(X_{1},X_{2})=2A(X)S(X_{1},X_{2})+A(X_{1})S(X,X_{2})+A(X_{2})S(X_{1},X)$

holds $\forall~{}X,X_{i}\in\chi(M)$ and a non-zero 1-form $A$ on $\\{x\in
M:S_{x}\neq 0\\}$.
As a generalization of Chaki pseudosymmetric manifold and recurrent manifold,
in 1989 Tam$\acute{\mbox{a}}$ssy and Binh [85] introduced the notion of weakly
symmetric manifold. The manifold $M$ is said to be weakly symmetric by
Tam$\acute{\mbox{a}}$ssy and Binh [85] (briefly, $WS_{n}$) if

$\displaystyle\nabla_{X}R(X_{1},X_{2},X_{3},X_{4})=A(X)R(X_{1},X_{2},X_{3},X_{4})+B(X_{1})R(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle\hskip
56.9055pt+\overline{B}(X_{2})R(X_{1},X,X_{3},X_{4})+D(X_{3})R(X_{1},X_{2},X,X_{4})+\overline{D}(X_{4})R(X_{1},X_{2},X_{3},X)$

holds $\forall~{}X,X_{i}\in\chi(M)$ $(i=1,2,3,4)$ and some 1-forms
$A,B,\overline{B},D$ and $\overline{D}$ on $\\{x\in M:R_{x}\neq 0\\}$.
Similarly for other $(0,4)$-tensors we obtain various weak symmetry
structures, e.g., conformally weak symmetry (briefly, $CWS_{n}$), projectively
weak symmetry, concircularly weak symmetry, conharmonically weak symmetry etc.
It may be mentioned that every weakly symmetric manifold is Chaki
pseudosymmetric ([31], [61]). For the reduced defining condition of various
weak symmetry we refer the reader to see the recent investigation on
pseudosymmetric manifolds by Shaikh et al. [61] and also references therein.
Again in 1993 Tam$\acute{\mbox{a}}$ssy and Binh [86] introduced the notion of
weakly Ricci symmetric manifold. The manifold $M$ is said to be weakly Ricci
symmetric [86] (briefly, $WRS_{n}$) if

$(\nabla_{X}S)(X_{1},X_{2})=A(X)S(X_{1},X_{2})+B(X_{1})S(X,X_{2})+D(X_{2})S(X_{1},X)$

holds $\forall~{}X,X_{i}\in\chi(M)$ and some 1-forms $A,B,D$ on $\\{x\in
M:S_{x}\neq 0\\}$. Again replacing $S$ by some other $(0,2)$-tensors, we get
various weak symmetry structures. In [61] Shaikh et al. presented the reduced
form of the defining condition of weak symmetry for various $(0,2)$-tensors.
Weakly symmetric and weakly Ricci symmetric manifolds by Tamássy and Binh were
investigated by Shaikh and his coauthors (see, [10], [11], [36], [62], [63],
[64], [65], [66], [68], [69], [70], [71], [72], [80] and also references
therein). Recently in [74] Shaikh and Kundu studied the characterization of
warped product $WS_{n}$ and $WRS_{n}$.
For a $(0,4)$-tensor $H$ and a symmetric $(0,2)$-tensor $E$ we can define two
endomorphisms $X\wedge_{E}Y$ and $\mathscr{H}(X,Y)$ by ([15], [21], [33])

$\displaystyle(X\wedge_{E}Y)X_{1}=E(Y,X_{1})X-E(X,X_{1})Y\ \ \mbox{ and }\ \
\mathscr{H}(X,Y)X_{1}=\mathcal{H}(X,Y)X_{1}$

respectively, where $\mathcal{H}$ is the corresponding $(1,3)$-tensor of $H$.
We note that the corresponding endomorphism of $R$ is given by
$\mathscr{R}(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}$, where the square
bracket is the Lie bracket over $\chi(M)$.
For a $(0,k)$-tensor $T$, $k\geq 1$, a $(0,4)$-tensor $H$ and a symmetric
$(0,2)$-tensor $E$ we define two $(0,k+2)$-tensors $H\cdot T$, and $Q(E,T)$ by
([15], [21], [23], [33])

$\displaystyle(H\cdot T)(X_{1},\cdots,X_{k},X,Y)=(\mathcal{H}(X,Y)\cdot
T)(X_{1},\cdots,X_{k})$
$\displaystyle=-T(\mathcal{H}(X,Y)X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},\cdots,X_{k-1},\mathcal{H}(X,Y)X_{k}),$
$\displaystyle Q(E,T)(X_{1},\cdots,X_{k};X,Y)=((X\wedge_{E}Y)\cdot
T)(X_{1},\cdots,X_{k})$
$\displaystyle=-T((X\wedge_{E}Y)X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},\cdots,X_{k-1},(X\wedge_{E}Y)X_{k}).$

Putting in the above formulas $T=R,S,C,P,W,K$ and $E=g$ or $S$, we obtain the
tensors: $R\cdot R$, $R\cdot S$, $R\cdot C$, $R\cdot P$, $R\cdot W$, $R\cdot
K$, $C\cdot R$, $C\cdot S$, $C\cdot C$, $C\cdot P$, $C\cdot W$, $C\cdot K$,
$P\cdot R$, $P\cdot C$, $P\cdot P$, $P\cdot W$, $P\cdot K$, $Q(g,R)$,
$Q(g,S)$, $Q(g,C)$, $Q(g,P)$, $Q(g,W)$, $Q(g,K)$, $Q(S,R)$, $Q(S,C)$,
$Q(S,P)$, $Q(S,W)$, $Q(S,K)$ etc. The tensor $Q(E,T)$ is called the Tachibana
tensor of the tensors $E$ and $T$, or the Tachibana tensor for short [23].
If the manifold $M$ satisfies the condition $R\cdot R=0$ (resp., $R\cdot S=0$,
$R\cdot C=0$, $R\cdot W=0$, $R\cdot P=0$, $R\cdot K=0$), then it is called
semisymmetric ([7], [82]) (briefly, $SS_{n}$) (resp., Ricci semisymmetric
(briefly, $RSS_{n}$), conformally semisymmetric (briefly, $CSS_{n}$),
projectively semisymmetric, concircularly semisymmetric, conharmonically
semisymmetric).
The manifold $M$ is said to be pseudosymmetric in sense of Deszcz or simply
Deszcz pseudosymmetric (resp., Ricci pseudosymmetric) [24] if the condition

(2.2) $R\cdot R=L_{R}\,Q(g,R)\ \ (\mbox{resp.,}R\cdot S=L_{S}\,Q(g,S))$

holds on $U_{R}=\\{x\in M:\left(R-\frac{\kappa}{n(n-1)}G\right)_{x}\neq 0\\}$
(resp., $U_{S}=\\{x\in M:\left(S-\frac{r}{n}\,g\right)_{x}\neq 0\\}$) for a
function $L_{R}$ (resp., $L_{S}$) on $U_{R}$ (resp., $U_{S}$).
The manifold $M$, $n\geq 4$, is said to be a manifold with pseudosymmetric
Weyl conformal curvature tensor ([17], [25]) (resp., pseudosymmetric Weyl
projective curvature tensor) if

(2.3) $C\cdot C=L_{C}\,Q(g,C)\ \ (\mbox{resp., }P\cdot P=L_{P}\,Q(g,P))$

on $U_{C}=\\{x\in M:C\neq 0\ \mbox{at}\ x\\}$ (resp., $U_{P}=\\{x\in
M:Q(g,P)\neq 0\ \mbox{at}\ x\\}$), where $L_{C}$ (resp., $L_{P}$) is a
function on $U_{C}$ (resp., $U_{P}$). We note that $U_{C}\subset U_{R}$.
Again $M$ is said to be Ricci-generalized pseudosymmetric ([13], [14]) if at
every point of $M$, the tensor $R\cdot R$ and the Tachibana tensor $Q(S,R)$
are linearly dependent. Hence $M$ is Ricci-generalized pseudosymmetric if and
only if

(2.4) $R\cdot R=L\,Q(S,R)$

holds on $\\{x\in M:Q(S,R)\neq 0\ \mbox{at}\ x\\}$, where $L$ is some function
on this set. An important subclass of Ricci-generalized pseudosymmetric
manifolds is formed by the manifolds realizing the condition ([13], [25])

(2.5) $R\cdot R=Q(S,R).$

The manifold $M$, $n\geq 4$, is said to be a manifold with Ricci generalized
pseudosymmetric Weyl conformal curvature tensor ([17], [25]) (resp., Ricci
generalized pseudosymmetric Weyl projective curvature tensor) if

(2.6) $C\cdot C=L_{1}\,Q(S,C)\ \ (\mbox{resp., }P\cdot P=L_{2}\,Q(S,P))$

on $U_{1}=\\{x\in M:Q(S,C)\neq 0\ \mbox{at}\ x\\}$ (resp., $U_{2}=\\{x\in
M:Q(S,P)\neq 0\ \mbox{at}\ x\\}$), where $L_{1}$ (resp., $L_{2}$) is a
function on $U_{1}$ (resp., $U_{2}$).
A symmetric $(0,2)$-tensor $E$ on $M$ is called Riemann compatible or simply
$R$-compatible ([41], [42]) if

$R(\mathcal{E}X_{1},X,X_{2},X_{3})+R(\mathcal{E}X_{2},X,X_{3},X_{1})+R(\mathcal{E}X_{3},X,X_{1},X_{2})=0$

holds $\forall~{}X,X_{1},X_{2},X_{3}\in\chi(M)$, where $\mathcal{E}$ is the
endomorphism on $\chi(M)$ defined as
$g(\mathcal{E}X_{1},X_{2})=E(X_{1},X_{2})$.
Again a vector field $Y$ is called $R$-compatible (also known as Riemann
compatible [43]) if

$\Pi(X_{1})R(Y,X,X_{2},X_{3})+\Pi(X_{2})R(Y,X,X_{3},X_{1})+\Pi(X_{3})R(Y,X,X_{1},X_{2})=0$

holds $\forall~{}X,X_{1},X_{2},X_{3}\in\chi(M)$, where $\Pi$ is the
corresponding 1-form of $Y$, i.e., $\Pi(X)=g(X,Y)$. We note that if $Y$ is
Riemann compatible, then $\Pi\otimes\Pi$ is a Riemann compatible tensor.
Similarly for other curvature tensors we have conformal compatibility (also
known as Weyl compatibility see, [22] and [43]), concircular compatibility,
conharmonic compatibility for both a vector and a $(0,2)$-tensor.
Recently Mantica and Suh ([44], [45], [46] and [47]) obtained a necessary and
sufficient condition for the recurrency of the curvature $2$-forms
$\Omega_{(R)l}^{m}=R_{jkl}^{m}dx^{j}\wedge dx^{k}$ ([4], [40]) and Ricci
1-forms $\Lambda_{(S)l}=S_{lm}dx^{m}$, where $\wedge$ indicates the exterior
product. They showed that $\Omega_{(R)l}^{m}$ are recurrent (i.e.,
$\mathcal{D}\Omega_{(R)l}^{m}=A\wedge\Omega_{(R)l}^{m}$, $\mathcal{D}$ is the
exterior derivative and $A$ is the associated $1$-form) if and only if

$\displaystyle(\nabla_{X_{1}}R)(X_{2},X_{3},X,Y)+(\nabla_{X_{2}}R)(X_{3},X_{1},X,Y)+(\nabla_{X_{3}}R)(X_{1},X_{2},X,Y)=$
$\displaystyle\hskip
72.26999ptA(X_{1})R(X_{2},X_{3},X,Y)+A(X_{2})R(X_{3},X_{1},X,Y)+A(X_{3})R(X_{1},X_{2},X,Y)$

and $\Lambda_{(S)l}$ are recurrent (i.e.,
$\mathcal{D}\Lambda_{(S)l}=A\wedge\Lambda_{(S)l}$) if and only if

$(\nabla_{X_{1}}S)(X_{2},X)-(\nabla_{X_{2}}S)(X_{1},X)=A(X_{1})S(X_{2},X)-A(X_{2})S(X_{1},X)$

for an 1-form $A$. If on a manifold the curvature 2-forms are recurrent for
$R$ and $C$, then we denote such a manifold by $K_{n}(2)$ and $CK_{n}(2)$
respectively. Again if the Ricci 1-forms are recurrent then it is denoted by
$RK_{n}(1)$.

##  3\. Existence of $HGK_{n}$

Let $M$ be an open connected subset of $\mathbb{R}^{4}$ such that
$x^{1},x^{2},x^{3},x^{4}>0$ endowed with the Lorentzian metric

(3.1)
$ds^{2}=g_{ij}dx^{i}dx^{j}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}+e^{x^{1}}(dx^{4})^{2},\
\ i,j=1,2,3,4.$

Then the non-zero components of the Christoffel symbols of second kind (upto
symmetry) are given by:

$\Gamma^{2}_{11}=-\Gamma^{3}_{11}=\Gamma^{2}_{13}=\frac{e^{x^{1}+x^{3}}}{2},\
\Gamma^{4}_{14}=\frac{1}{2},\ \Gamma^{2}_{44}=-\frac{e^{x^{1}}}{2}.$

The non-zero components of curvature tensor and Ricci tensor (upto symmetry)
are given by:

(3.2) $R_{1313}=-\frac{1}{2}e^{x^{1}+x^{3}},\ \ \
R_{1414}=-\frac{e^{x^{1}}}{4},\ \ \ \ \
S_{11}=\frac{1}{4}(1+2e^{x^{1}+x^{3}}).$

The scalar curvature of this metric is given by $r=0$. Again the non-zero
components $R_{hijk,l}$ and $S_{ij,l}$ of the covariant derivatives of
curvature tensor and Ricci tensor (upto symmetry) are given by:

(3.3) $R_{1313,1}=-\frac{e^{x^{1}+x^{3}}}{2}=R_{1313,3},\ \ \ \
S_{11,1}=S_{13,3}=-\frac{e^{x^{1}+x^{3}}}{2}.$

The non-zero components of the conformal curvature tensor and its covariant
derivative (upto symmetry) are given by:

(3.4) $\left\\{\begin{array}[]{c}C_{1313}=\frac{1}{8}(1-2e^{x^{1}+x^{3}}),\ \
\ C_{1414}=-\frac{1}{8}e^{x^{1}}(1-2e^{x^{1}+x^{3}}),\vspace{0.1in}\\\
C_{1313,1}=C_{1313,3}=-\frac{e^{x^{1}+x^{3}}}{4},\ \ \
C_{1414,1}=C_{1414,3}=\frac{e^{2x^{1}+x^{3}}}{4}.\end{array}\right.$

The non-zero components of the conharmonic curvature tensor and its covariant
derivative (upto symmetry) are given by:

(3.5) $\left\\{\begin{array}[]{c}K_{1313}=\frac{1}{8}(1-2e^{x^{1}+x^{3}}),\ \
\ K_{1414}=-\frac{1}{8}e^{x^{1}}(1-2e^{x^{1}+x^{3}}),\vspace{0.1in}\\\
K_{1313,1}=K_{1313,3}=-\frac{e^{x^{1}+x^{3}}}{4},\ \ \
K_{1414,1}=K_{1414,3}=\frac{e^{2x^{1}+x^{3}}}{4}.\end{array}\right.$

The non-zero components of the projective curvature tensor and its covariant
derivative (upto symmetry) are given by:

(3.6)
$\left\\{\begin{array}[]{c}P_{1211}=\frac{1}{12}\left(2e^{x^{1}+x^{1}}+1\right),\
\ P_{1313}=\frac{1}{12}\left(1-4e^{x^{1}+x^{1}}\right),\ \
P_{1331}=\frac{1}{2}e^{x^{1}+x^{1}}\\\
P_{1414}=\frac{1}{6}e^{x^{1}}\left(e^{x^{1}+x^{1}}-1\right),\ \ \
P_{1441}=\frac{e^{x^{1}}}{4}\\\
\vspace{0.1in}P_{1211,1}=P_{1211,3}=-P_{1313,1}=-P_{1313,3}=P_{1331,1}=P_{1331,3}=\frac{1}{2}e^{x^{1}+x^{3}}\\\
P_{1414,1}=P_{1414,3}=\frac{1}{6}e^{2x^{1}+x^{3}}.$$\end{array}\right.$

The non-zero components of the concircular curvature tensor and its covariant
derivative (upto symmetry) are given by:

(3.7) $W_{1313}=-\frac{1}{2}e^{x^{1}+x^{3}},\ \ W_{1414}=-\frac{e^{x^{1}}}{4}\
\mbox{ and }\ W_{1313,1}=W_{1313,3}=-\frac{1}{2}e^{x^{1}+x^{3}}.$

In view of (3.2)–(3.7), it can be easily shown that the tensors $R\cdot R$,
$C\cdot R$, $P\cdot R$, $P\cdot S$, $P\cdot C$, $P\cdot Z$, $P\cdot K$,
$W\cdot R$ and $K\cdot R$ are identically zero. Thus from [75] we can conclude
that $R\cdot H$, $C\cdot H$, $W\cdot H$ and $K\cdot H$ are all identically
zero for $H=C,W$ and $K$.
The non-zero components of $P\cdot P$ and $Q(S,P)$ are (upto symmetry) given
below:

(3.8) $(P\cdot P)_{131131}=\frac{1}{144}\left(1+2e^{x^{1}+x^{3}}\right)^{2},\
\ \ (P\cdot
P)_{141141}=\frac{1}{144}e^{x^{1}}\left(1+2e^{x^{1}+x^{3}}\right)^{2}.$ (3.9)
$Q(S,P)_{141114}=e^{x^{1}}Q(S,P)_{131113}=\frac{e^{x^{1}}}{48}(1+2e^{x^{1}+x^{3}})^{2}.$

In terms of local coordinates the defining condition of a $HGK_{n}$ can be
written as

(3.10)
$R_{hijk,\,l}=A_{l}R_{hijk}+B_{l}[S_{hk}g_{ij}+S_{ij}g_{hk}-S_{hj}g_{ik}-g_{hj}S_{ik}],\
\ 1\leq h,i,j,k,l\leq n.$

Then by virtue of (3.1), (3.2), (3.3) and (3.10) it is easy to check that the
manifold is a $HGK_{4}$ with associated 1-forms

(3.11)
$\left\\{\begin{array}[]{ccc}&A_{i}=\left\\{\begin{array}[]{ccc}&-\frac{2e^{x^{1}+x^{3}}}{1-2e^{x^{1}+x^{3}}}&\
\ \ \ \mbox{for}\ \ \ i=1,3\\\ &0&\ \ \ \
\mbox{otherwise,}\end{array}\right.&\\\ \\\
&B_{i}=\left\\{\begin{array}[]{ccc}&\frac{2e^{x^{1}+x^{3}}}{1-4e^{2x^{1}+2x^{3}}}&\
\ \ \ \mbox{for}\ \ \ i=1,3\\\ &0&\ \ \ \
\mbox{otherwise.}\end{array}\right.&\end{array}\right.$

If we consider the 1-form $\eta=\\{1,0,0,0\\}$ and the scalar
$\beta=\frac{1}{4}(1+2e^{x^{1}+x^{3}})$ then from (2.1) and (3.2) it follows
that the manifold satisfies $S=\beta\eta\otimes\eta$. Thus the manifold is
Ricci simple.
Now it is easy to check that $\eta$ is parallel, i.e., $\nabla\eta=0$, hence
from Proposition 2.1, the manifold is Ricci recurrent. Also by virtue of (3.2)
and (3.3), we can easily check that the manifold is Ricci recurrent with the
associated 1-form

$A_{i}=\left\\{\begin{array}[]{ccc}&\frac{2e^{x^{1}+x^{3}}}{1+2e^{x^{1}+x^{3}}}&\
\ \ \ \mbox{for}\ \ \ i=1,3\\\ &0&\ \ \ \ \mbox{otherwise.}\end{array}\right.$

Now from (3.5) it follows that the manifold is conharmonically recurrent with
the associated 1-form

$A_{i}=\left\\{\begin{array}[]{ccc}&-\frac{2e^{x^{1}+x^{3}}}{1-2e^{x^{1}+x^{3}}}&\
\ \ \ \mbox{for}\ \ \ i=1,3\\\ &0&\ \ \ \ \mbox{otherwise.}\end{array}\right.$

Now from the values of $R$ it can be easily checked that $M$ satisfies the
condition

$\Pi(X_{1})R(X_{2},X_{3},X,Y)+\Pi(X_{2})R(X_{3},X_{1},X,Y)+\Pi(X_{3})R(X_{1},X_{2},X,Y)=0$

for the 1-form

$\Pi_{i}=\left\\{\begin{array}[]{ccc}&\frac{2e^{x^{1}+x^{3}}}{1+2e^{x^{1}+x^{3}}}&\
\ \ \ \mbox{for}\ \ \ i=1\\\ &0&\ \ \ \ \mbox{otherwise.}\end{array}\right.$

Thus from (2), it follows by virtue of second Bianchi identity that the
curvature 2-forms $\Omega_{(R)l}^{m}$ are recurrent.
The general form of Riemann compatible tensor $E$ with local components
$E_{ij}$, of $M$ can be easily be evaluated as

$\left(\begin{array}[]{cccc}E_{11}&E_{12}&E_{13}&E_{14}\\\ E_{21}&0&0&0\\\
E_{31}&0&E_{33}&2e^{x^{1}+x^{3}}E_{43}\\\
E_{41}&0&E_{43}&E_{44},\end{array}\right)$

where $E_{ij}$’s are arbitrary. Again the general form of Riemann compatible
vector $Y$ is
$\left\\{0,a_{1},0,a_{4}e^{-x^{1}}\right\\}$ or
$\left\\{{a_{2},-a_{2}e^{x^{1}+x^{3}},a_{3},0}\right\\}.$
Similarly we can get the conformal compatible, concircular compatible and
conharmonic compatible tensors respectively given in the following form:

$\displaystyle\left(\begin{array}[]{cccc}E_{11}&E_{12}&E_{13}&E_{14}\\\
E_{21}&0&0&0\\\ E_{31}&0&E_{33}&E_{34}\\\
E_{41}&0&-E_{34}&E_{44}\end{array}\right),\left(\begin{array}[]{cccc}E_{11}&E_{12}&E_{13}&E_{14}\\\
E_{21}&0&0&0\\\ E_{31}&0&E_{33}&2e^{x^{1}+x^{3}}E_{43}\\\
E_{41}&0&E_{43}&E_{44}\end{array}\right)\mbox{and}\left(\begin{array}[]{cccc}E_{11}&E_{12}&E_{13}&E_{14}\\\
E_{21}&0&0&0\\\ E_{31}&0&E_{33}&E_{34}\\\
E_{41}&0&-E_{34}&E_{44}\end{array}\right).$

Again the general form of conformal (resp., concircular and conharmonic)
compatible vector $Y$ is $\left\\{0,a_{1},0,a_{4}e^{-x^{1}}\right\\}$ or
$\left\\{{a_{2},-a_{2}e^{x^{1}+x^{3}},a_{3},0}\right\\}.$

######  Remark 3.1.

From Lemma 5.6 of [75] it follows that if a semi-Riemannian manifold $M$ is
Ricci recurrent and also conformally recurrent with same 1-form of recurrence,
then $M$ is recurrent. We note that the above manifold is Ricci recurrent and
also conformally recurrent but of different 1-forms of recurrence and it is
not recurrent.

Now we can state the following:

######  Proposition 3.1.

Let $(M^{4},g)$ be a semi-Riemannian manifold equipped with the metric (3.1).
Then $(M^{4},g)$ is:
(i) Ricci simple and hence a special quasi Einstein manifold,
(ii) manifold of constant scalar curvature,
(iii) Ricci recurrent and hence weakly Ricci symmetric and the Ricci 1-forms
are recurrent $($see [61] and [47]$)$,
(iv) conharmonically recurrent and hence conformally recurrent $($see
Corollary 6.2 of [75]$)$ as well as conharmonically and conformally weakly
symmetric, and the curvature 2-forms for $C$ and $K$ are recurrent,
(v) hyper-generalized recurrent and hence generalized Ricci recurrent,
(vi) semisymmetric and hence fulfills the semisymmetry conditions for $S$,
$C$, $P$, $W$ and $K$,
(vii) weakly Ricci symmetric and weakly conharmonic symmetric such that all
the associated 1-forms are different $($This result proves the non uniqueness
of associated 1-forms of weak symmetry structures, see Section 3 of [61]$)$,
(viii) recurrent curvature 2-form, and
(ix) fulfills the condition $P\cdot P=-\frac{1}{3}Q(S,P)$, i.e, Ricci
generalized pseudosymmetric Weyl projective curvature tensor.

From the above values of various tensors, we can state the following:

######  Proposition 3.2.

The 4-dimensional semi-Riemannian manifold $M$ equipped with the metric given
in (3.1) is not any one the following:
(i) Einstein, (ii) class $\mathcal{A}$ or $\mathcal{B}$, (iii) pseudosymmetric
in sense of Chaki for $R$, $C$, $P$, $W$ and $K$ (iv) pseudo Ricci symmetric,
(v) quasi generalized recurrent, (vi) weakly generalized recurrent, (vii)
$P\cdot P=0$, (viii) Codazzi type Ricci tensor, (ix) cyclic Ricci parallel,
(x) weakly symmetric for $R$ and $Z$.

It is well known that recurrent structure implies Ricci recurrent, conformally
recurrent and also conharmonically recurrent but the converse is not true in
each case. Again Chaki pseudosymmetry for $R$, $S$, $C$, $P$, $W$ and $K$
implies weak symmetry for $R$, $S$, $C$, $P$, $W$ and $K$ respectively but the
converse cases are not true. Now from above two propositions we can state the
following:

######  Theorem 3.1.

Let $(M^{4},g)$ be a semi-Riemannian manifold equipped with the metric (3.1).
Then $(M^{4},g)$ is
(i) not locally symmetric but semisymmetric,
(ii) not weakly symmetric but weakly Ricci symmetric as well as conformally
weakly symmetric,
(iii) not recurrent but Ricci recurrent as well as conformally recurrent,
(iv) not recurrent but hyper-generalized recurrent,
(v) neither class $\mathcal{A}$ nor $\mathcal{B}$ but of constant scalar
curvature,
(vi) not Einstein but Ricci simple i.e., special quasi-Einstein,
(vii) not recurrent but the curvature 2-forms are recurrent.
(viii) does not satisfy $P\cdot P=0$ but fulfills the condition $P\cdot
P=LQ(S,P)$.

######  Remark 3.2.

From above theorem we get an affirmative answer of Q. 1 and we can state the
following:
(i) semisymmetry is a proper generalization of local symmetry and the result
is also true for the case $S$, $C$, $P$, $W$ and $K$,
(ii) Ricci recurrency and conharmonic recurrency are both proper
generalizations of recurrency,
(iii) weak Ricci symmetry is a proper generalization of pseudo Ricci symmetry
and the result is also true for conharmonic and concircular curvature tensors,
(iv) the classes $\mathcal{A}$ and $\mathcal{B}$ properly lie between Ricci
symmetry and manifold of constant scalar curvature,
(v) hyper-generalized recurrency is a proper generalization of recurrency,
(vi) recurrency of curvature 2-forms is a proper generalization of recurrency.

The above remark can be stated in a diagram as shown in Figure 3.1, where the
boxes denote the corresponding class of manifolds and the ‘implication’ signs
denote that the later one is a generalization of the previous. The above
example ensures the properness of some of them, which are indicated by the
dark filled ones.

Figure 3.1. Some generalizations of local symmetry

Again, if we change the signature of the metric (3.1) as

$ds^{2}=g_{ij}dx^{i}dx^{j}=-e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}+e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}-(dx^{3})^{2}-e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=-e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}-(dx^{3})^{2}-e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}-(dx^{3})^{2}+e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=-e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}-(dx^{3})^{2}+e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=-e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}-e^{x^{1}}(dx^{4})^{2},$
$ds^{2}=g_{ij}dx^{i}dx^{j}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}-e^{x^{1}}(dx^{4})^{2},$

$i,j=1,2,3,4$, then it can be easily shown that the geometric properties
presented in Theorem 3.1 remain unchanged for the above metrics. We note that
the first three of the above metrics are Lorentzian and remaining are semi-
Riemannian.

######  Remark 3.3.

Thus we get the existence of a new class of semi-Riemannian manifolds having
the geometric properties presented in Theorem 3.1. It is similar to show that
the following metrics (also with different signatures) belong to this class:

(3.13)
$ds^{2}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}+f(x^{1})(dx^{4})^{2},$
(3.14)
$ds^{2}=x^{1}x^{3}(dx^{1})^{2}+x^{1}(dx^{2})^{2}+2dx^{1}dx^{3}+f(x^{1})(dx^{4})^{2},$
(3.15)
$ds^{2}=e^{x^{1}+x^{3}}(dx^{1})^{2}+2dx^{1}dx^{2}+(dx^{3})^{2}+f(x^{1})(dx^{4})^{2}+f(x^{1})\delta_{ab}dx^{a}dx^{b},$
(3.16)
$ds^{2}=x^{1}x^{3}(dx^{1})^{2}+x^{1}(dx^{2})^{2}+2dx^{1}dx^{3}+f(x^{1})(dx^{4})^{2}+f(x^{1})\delta_{ab}dx^{a}dx^{b}$

where $f$ is a smooth function and $\delta_{ab}$ denotes the Kronecker delta,
$5\leq a,b\leq n$.

Conclusion: From the above results and discussion we get the existence of a
new class of semi-Riemannian manifolds which are hypergeneralized recurrent,
Ricci recurrent, conharmonically recurrent, conformally recurrent, manifold of
recurrent curvature 2-forms, Ricci simple (thus special quasi Einstein),
weakly Ricci symmetric, conformally weakly symmetric, conharmonically weakly
symmetric, semisymmetric, Ricci semisymmetry and conformally semisymmetric and
fulfills the pseudosymmetric type condition $P\cdot P=-\frac{1}{3}Q(S,P)$ but
do not satisfy any one of the conditions (i)-(x) of Proposition 3.2. Hence the
existence of a generalized class of recurrent manifolds, viz., hyper
generalized recurrent manifolds is ensured by the metric.

Acknowledgement. We have made all the calculations by Wolfram Mathematica. The
last named author gratefully acknowledges to CSIR, New Delhi, India (File No.
09/025 (0194)/2010-EMR-I) for the financial support.

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Software for full-color 3D reconstruction of the biological tissues internal
structure

Khoperskov A.V.1[0000-0003-0149-7947], Kovalev M.E.1[0000-0001-5136-4048],
Astakhov A.S.1[0000-0002-9905-568X], Novochadov V.V.2[0000-0001-6317-7418],
Terpilovskiy A.A.3[0000-0001-8200-044X], Tiras K.P.4[0000-0002-1853-8285],
Malanin D.A.5[0000-0002-8065-6506]

1 Institute of Mathematics and Information Technologies, Volgograd State
University, Volgograd, Russia

2 Institute of Natural Sciences, Volgograd State University, Volgograd, Russia

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
3 The laboratory of virtual biology, Ltd., Moscow, Russia

4 Institute of Theoretical and Experimental Biophysics of RAS, Moscow, Russia

5 Volgograd State Medical University, Volgograd, Russia

Abstract. A software for processing sets of full-color images of biological
tissue histological sections is developed. We used histological sections
obtained by the method of high-precision layer-by-layer grinding of frozen
biological tissues. The software allows restoring the image of the tissue for
an arbitrary cross-section of the tissue sample. Thus, our method is designed
to create a full-color 3D reconstruction of the biological tissue structure.
The resolution of 3D reconstruction is determined by the quality of the
initial histological sections. The newly developed technology available to us
provides a resolution of up to 5 - 10 $\displaystyle\mu$m in three dimensions.

Keywords: Scientific software, Computer graphics, 3D reconstruction,
Biological tissues, Image processing.

Introduction

Different approaches and methods of 3D-visualization of biological tissues are
discussed depending on the scientific goals and objectives of practical
applications [11, 12, 1314]. A significant number of software complexes,
libraries and specialized systems of scientific and technological
visualization for three-dimensional digital biomedicine have been developed
[67, 89, 1112, 2324, 2425]. Approaches using medical X-ray tomography are very
effective for 3D-reconstruction of tissues/organs [78]. We can also highlight
some areas that are associated with the localization of implanted biomaterials
[78, 2526], the colonoscopy, the ultrasonic sounding [89], the stereoscopic
fluorescence imaging, the multispectral magnetic resonance image analysis
[1516], the single photon emission computed tomography (CT) [1011], the
electron tomography [56], the use of combined methods [89]. The transition
begins from micro-CT to nano-CT [78, 1920].

This work is addressed to the anatomical or destructive tomography (biotomy)
approach [1617, 2223]. A newly developed method is based on making of a set of
high-quality images of biological tissue histological sections using the high-
precision grinding of a pre-frozen biological sample [2122].

The advantages of the proposed approach:

  1. 1.

A high-quality photography offering all the benefits of the raster graphics.

  2. 2.

A realistic color rendering.

  3. 3.

Very high accuracy up to several microns.

  4. 4.

Absence of "screening interference" in contrast to non-destructive methods
[2526].

  5. 5.

High speed access to information about individual fragments of the tissue.

The main disadvantage of the proposed method is principle impossibility of
living tissue usage due to its physical destruction during the process of
preparation of a set of images slices. Another disadvantage of the method
associates with utilization of digital raster photographs. 3D modeling can
improve the efficiency of various solving aims tasks, including
endoprosthetics [45, 78, 910, 1415, 2526, 2627].

Making of an images set of histological sections

Figure 1: The sequence of steps for making of an images set of biological
tissue histological sections

The process of producing of images set includes the following steps (Fig. 11):

  1. 1.

Selection of a biological object in accordance with the objectives of the
study.

  2. 2.

A formation of the original sample by fixing a biological object by a casting
material in a three-dimensional mold.

  3. 3.

The original sample of the biological object is rapidly frozen in the form of
a rectangular parallelepiped. The fixation of the biological object is ensured
by at cryogenic temperatures (T $\displaystyle\leq-72^{\circ}$C), after that
it is filled by the casting material (at T $\displaystyle\leq-25^{\circ}$C)
[1819].

  4. 4.

The sample is placed into a special device for layer-by-layer grinding.

  5. 5.

An optical system adjustment is required due to the sample thickness change as
a result of layer-by-layer removing.

  6. 6.

Visual analysis of the images allows distinguishing photos with a blurred
focus, highly modified illumination and other optical artifacts (removing
images of poor quality if it is necessary).

As a result, we have an ordered snapshot sequence of biological tissue slices
with step  $\displaystyle\Delta$z, whose resolution determined by sample size
and used object-glass. The  $\displaystyle\Delta$x and  $\displaystyle\Delta$y
resolution of the equipment available at The Laboratory of Virtual Biology,
Ltd is 6-20 $\displaystyle\mu$m, while the  $\displaystyle\Delta$z resolution
is 5 - 20 $\displaystyle\mu$m.

Creation of $\displaystyle\Theta$-sections and software development

Let us introduce a $\displaystyle\Theta$-rectangle ("theta-rectangle") to
denote a rectangle inscribed into the polygon of the sample section. Various
ways of $\displaystyle\Theta$-rectangle specification in a triangular section
are shown in Fig.22.

Figure 2: The examples of the $\displaystyle\Theta$-rectangles in a triangular
section

Hereafter the RGB color system is applied. Each element of the mapping color
table P is considered like a vector in the three-dimensional space. Thereby,
within the closure region of the sample space the vector field
$\displaystyle\vec{\Omega}\left(x,y,z\right)=\\{R,G,B\\}$ is a set that
converts it into a color space. To construct an image on the sample slice
plane, the -rectangle of this section have to be divided into pixels. Each
pixel is associated with a color depending on its spatial position in a vector
field $\displaystyle\vec{\Omega}$.

A set of slices images is the initial data

$z_{k}=\left(k-1\right)\Delta z,$ (1)

where the k-th image number (k=1,…,M) is determined by the z-coordinate of the
histological section.

The sample vertex in the upper left corner of its frontal face relatively to
the camera location during the record is taken as origin. Let us Consider a
numbered set of raster images (1) of size $\displaystyle P_{x}\times P_{y}$,
where $\displaystyle P_{x}$ and $\displaystyle P_{x}$ are the numbers of
pixels for the two corresponding sides. Such an initial set of images is also
called a digital set of base slices.

To reconstruct an object in the form of a three-dimensional model, a method
for color interpolation in the space between each adjoining pair of images in
the plains $\displaystyle\textit{z}_{\textit{k}}$ and
$\displaystyle\textit{z}_{\textit{k+1}}$ should be specified. The entire set
of initial images is considered as a three-dimensional matrix
$\displaystyle\widehat{P}=\left(i,j,k\right)$ , whose elements are individual
pixels of images from the set. Their addressing is determined by the number of
the k-th layer (a sequence number of a photo), the number of the i-th row and
the number of the j-th column in the matrix of pixels of the k-th photo. We
have used discrete functions defined on a three-dimensional grid
$\displaystyle\textit{(x}_{\textit{i}}\textit{,y}_{\textit{j}}\textit{,z}_{\textit{k}}\textit{)}$.

The statement that the vector is orthogonal to the normal vector to the plane
described by the equation $\displaystyle\textit{Ax + By + Cz + D = }0$ is
equivalent tantamount to the following expression:

$A\xi_{x}+C\xi_{z}=0.$ (2)

The vector  $\displaystyle\vec{\eta}$ obeys the conditions of orthogonality to
the vector  $\displaystyle\xi$ and the normal to the section plane. The latter
means that its direction may be computed as the vector product of the vector
$\displaystyle\xi$ by the normal vector to the section plane:

$\vec{\eta}=\left[\xi,n\right]=det\begin{pmatrix}e_{x}~{}e_{y}~{}e_{z}\\\
\xi_{x}~{}\xi_{y}~{}\xi_{z}\\\ A~{}B~{}C\end{pmatrix},$ (3)

where n is the normal vector to the plane of the section, $\displaystyle
e_{x}$, $\displaystyle e_{y}$, $\displaystyle e_{z}$ are the unit vectors
along the positive directions of the x-, y\- and z-axes respectively in the
reference system associated with the sample.

Finally, taking into account the fact that the vectors $\displaystyle\xi$ and
$\displaystyle\vec{\eta}$ have a unit length, we obtain two more equations:

$\sqrt[]{\xi_{x}^{2}+~{}\xi_{z}^{2}}=1,$ (4)
$\sqrt[]{\vec{\eta}_{x}^{2}+\vec{\eta}_{y}^{2}+\vec{\eta}_{z}^{2}}=1.$ (5)

From equations (2-5), we can calculate the coordinate values for the vectors
$\displaystyle\xi$ and  $\displaystyle\vec{\eta}$:

$\xi=~{}\left(\frac{C}{\sqrt[]{A^{2}+C^{2}}},~{}0,-\frac{A}{A^{2}+C^{2}}\right),$
(6)
$\vec{\eta}=~{}\left(-\frac{AB}{L},~{}\frac{A^{2}+C^{2}}{L},-\frac{BC}{L}\right),$
(7) $L=\sqrt[]{A^{2}B^{2}+\left(A^{2}+C^{2}\right)^{2}+B^{2}C^{2}}.$ (8)

To solve the problem of the long time required a data reading from the hard
disk, we choose a special basis on the plane of the section. We denote by p, q
the unit vectors along the directions of the abscissa and ordinate axes,
respectively, in the reference system associated with the section plane. Thus,
the vector p is orthogonal to the unit vector directed along the applicate
axis in the reference system associated with the sample:

$p_{z}=0,$ (9)

where $\displaystyle p_{z}$ is the projection of the vector $\displaystyle p$
onto the given axis.

Moreover, it is required adding equations denoting the orthogonality of the
vector p to the normal vector to the plane of the section, and the
orthogonality of the vector q to both of them, as well as the condition that
the length of the vectors p and q is equal to one. Accounting for all the
listed above conditions we obtain the following equations:

$Ap_{x}+Bp_{y}=0,$ (10) $q=det\begin{pmatrix}e_{x}~{}e_{y}~{}e_{z}\\\
p_{x}~{}p_{y}~{}p_{z}\\\ A~{}~{}B~{}~{}C\end{pmatrix},$ (11)
$\sqrt[]{p_{x}^{2}+p_{y}^{2}=1},$ (12)
$\sqrt[]{q_{x}^{2}+q_{y}^{2}+q_{z}^{2}=1}.$ (13)

From the equations (9-13), we can determine the values of the coordinates of
the basis vectors in the reference system associated with the sample:

$p=\left(-\frac{B}{\sqrt[]{A^{2}+B^{2}}},~{}\frac{A}{\sqrt[]{A^{2}+B^{2}}},0\right),$
(14) $q=\left(-\frac{AC}{M},-~{}\frac{BC}{M},\frac{A^{2}+B^{2}}{M}\right),$
(15) $M=\sqrt[]{A^{2}C^{2}+B^{2}C^{2}+\left(A^{2}+B^{2}\right)}.$ (16)

The equations (14) and (15) specify the directions of the basis vectors of
such reference system, associated with the cross-section plane, for which the
natural order of pixel bypass would guarantee the optimal number of calls to
information recorded on the hard disk.

Examples of images reconstruction in arbitrary sections

A human knee-joint (20 $\displaystyle\times$ 10 $\displaystyle\times$ 10 cm in
20 $\displaystyle\mu$m steps) and a rat knee-joints (with a pixel size of 8
$\displaystyle\times$ 8 $\displaystyle\mu$m and a pitch of 8
$\displaystyle\mu$m) are considered as an examples of the section
constructions. The samples were obtains in compliance with all legal norms.
These sets could be easily identified by color of casting material. Green and
red filling correspond to the rat and human knee-joints, respectively.

Images of different shapes of sections are considered below. If the section
does not have the shape of a rectangle, then the image is supplemented with a
black background (no restriction on the background color) (figures 33, 44, 55
and 66).

Figure 3: Triangular sections Figure 4: Cross-sections with 4 angles Figure 5:
Cross-sections with 5 angles Figure 6: Cross-sections with 6 angles

Information system for modeling

The developed software package includes two main parts: programs with a
graphical user interface (GUI) and calculation modules (DLL) for modeling
user-defined slices. Microsoft Visual Studio 2012 comprising C
$\displaystyle\\#$ programming language with .NET framework 4.5 has been
utilized as the software development environment. The programming interface is
refined by API WPF. User selects a folder with input data, sets the geometric
parameters of the sample (length, width, height and step
$\displaystyle\Delta$z as seen from Fig. 77) and sections (three points along
which the plane will be built). The file names are determined by the section
number in the sequence.

Figure 7: Setting of an arbitrary slice

After the previously described actions with the help of calculation modules,
the program creates an object of the Cuboid class (with the specified length,
width and depth) and an object of the Plane class describing the frontal face
of the cuboid (the normal vector is co-directed with the OZ axis and applied
to the origin — (0, 0, 0)).

Beside the one slide calculation, series of slices can be constructed (for
example, series of images with a given step in the range 1-6 ∘). In the case
if images size vary, they should be enlarged to the same size. In order to do
this all the resulting slices are analyzed and the image with the largest
width is selected. The remaining images are enlarged to chosen size and
supplemented with a gray-blue background (Fig. 88 ). The resulting images are
marked with angle values. Using the images movie can be compiled (for example,
for XY- and XZ- sections (along the Z- and Y- axes, respectively)).

Figure 8: Postprocessed image

Conclusions

A software product has been developed allowing work with images of sections
obtained by the method of high-precision grinding. The method of obtaining
slices and the algorithm for their subsequent processing completely exclude
spatial deformation and may be characterized by minimal color distortion.

To test the software complex and demonstrate the results of section
construction, we used samples of human and rat knee-joints. The virtual model
development of a human knee-joint may have practical application in new
technologies of restoration of articular surfaces at joint injuries [1617,
1718]. In addition, it may be applied for making of virtual simulators based
on the principles of augmented reality or for creation of 3D-printer models.
This method can be used for aims of medicine, veterinary medicine, zootechny
and related areas where MRI or histological reconstructions do not provide a
full-fledged three-dimensional view [23, 34, 910].

We can use this method to construct vector 3D models of samples that are
necessary for theoretical modeling physical processes in highly heterogeneous
biological tissues [1718, 2021]. Such 3D models allow to take into account the
fine structure of real tissues and changes in their physical parameters. The
joint use of image processing techniques and numerical modeling methods such
as computational fluid dynamics or heat transfer is a modern trend in the
development of medicine and applied biology [1213].

Acknowledgments. KAV and AAS are thankful to the Ministry of Education and
Science of the Russian Federation (project No. 2.852.2017/4.6). NVV thanks the
RFBR grant and Volgograd Region Administration (No. 15-47-02642).

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