Conjugate connections and statistical structures on almost Norden manifolds
Marta Teofilova

TL;DR
This paper explores the relationships between conjugate connections, Norden metrics, and almost complex structures on almost Norden manifolds, and investigates statistical structures within this geometric context.
Contribution
It introduces new conjugate connections related to Norden metrics and examines statistical structures on almost Norden manifolds, expanding understanding of their geometric properties.
Findings
Derived conjugate connections of Levi-Civita connections.
Analyzed statistical structures on almost Norden manifolds.
Established relations between conjugate connections and the almost complex structure.
Abstract
Relations between conjugate connections with respect to the pair of Norden metrics and to the almost complex structure on almost Norden manifolds are studied. Conjugate connections of the Levi-Civita connections induced by the Norden metrics are obtained. Statistical structures on almost Norden manifolds are considered.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Topics in Algebra · Geometric and Algebraic Topology
Conjugate connections and statistical structures on almost Norden
manifolds
Marta Teofilova
Abstract. Relations between conjugate connections with respect to the pair of Norden metrics and to the almost complex structure on almost Norden manifolds are studied. Conjugate connections of the Levi-Civita connections induced by the Norden metrics are obtained. Statistical structures on almost Norden manifolds are considered.
Key words: Norden metric, complex structure, conjugate connection, dual connection, complex conjugate connection, statistical manifold.
2010 Mathematics Subject Classification: Primary 53C15, 53C50; Secondary 32Q60.
Introduction
The concept of conjugate connections relative to a metric tensor field was originally introduced by A. P. Norden in the context of Weyl geometry [18]. Such linear connections were independently developed by H. Nagaoka and S. Amari [15] under the name dual connections and used by S. Lauritzen in the definition of statistical manifolds [12]. For more details on conjugate connections and their application to information theory, statistics and other fields see [2], [3], [7], [14], [17], [20].
Another kind of conjugate connections are those which are dual with respect to an invertible (1,1)-tensor field [1], [4]. Conjugate connections relative to an almost complex structure are studied by A. M. Blaga and M. Crasmareanu in [6]. Relations between conjugate connections with respect to a symplectic structure and to a complex structure on Kähler manifolds are investigated in [5]. Statistical structures and relations between conjugate connections on Hermitian manifolds are studied in [8], [16].
The main purpose of the present work111This work is partially supported by project FP17–FMI–008 of the Scientific Research Fund, University of Plovdiv Paisii Hilendarski, Bulgaria. is to study relations between both aforementioned types of conjugate connections on almost complex manifolds with Norden metric (B-metric). For the sake of brevity, such manifolds will be called almost Norden manifolds. These manifolds were introduced by A. P. Norden [19] and their geometry was studied for the first time by K. Gribachev, D. Mekerov and G. Djelepov [11] who termed them generalized B-manifolds.
Since on such manifolds, there exists a pair of Norden metrics, we can consider conjugate connections with respect to each of these metrical tensors and their relations to conjugate connections relative to the almost complex structure. Another aim of this work is to construct and study statistical structures on almost Norden manifolds.
The paper is organized as follows. In Section 1, we give some basic information about almost Norden manifolds and conjugate connections. In Section 2, we study the coincidence of conjugate connections with respect to the Norden metrics and the almost complex structure. The case of symmetric connections and completely symmetric connections is also investigated. In Section 3, we study curvature properties of the conjugate connections of the Levi-Civita connections induced by the pair of Norden metrics. In Section 4, we consider statistical structures on almost Norden manifolds by constructing families of linear connections with completely symmetric difference tensor and studying their curvature properties.
1. Preliminaries
1.1. Almost Norden manifolds
The triple is called an almost Norden manifold (almost complex manifold with Norden metric) if is a differentiable -dimensional manifold, is an almost complex structure, and is a pseudo Riemannian metric compatible with such that
[TABLE]
Here and further will stand for arbitrary vector fields on , i.e. elements in the Lie algebra , or vectors in the tangent space at an arbitrary point .
Equalities (1.1) imply which means that the tensor defined by
[TABLE]
is symmetric and is known as the associated (twin) metric of ( and are called a pair of twin metrics). This tensor also satisfies the Norden metric property, i.e. , i.e. is also an almost Norden manifold. Both metrics, and , are necessarily of neutral signature .
Let us denote by and the Levi-Civita connections of and , respectively. The tensor field defined by
[TABLE]
plays an important role in the geometry of almost Norden manifolds. It has the following properties
[TABLE]
Let be an arbitrary basis of , and be the components of the inverse matrix of with respect to this basis. The Lie 1-form associated with and its corresponding vector are given by
[TABLE]
A classification of the almost Norden manifolds with respect to the properties of is obtained by G. Ganchev and A. Borisov in [10]. This classification consists of eight classes: three basic classes (), their pairwise direct sums , the widest class and the class of the Kähler Norden manifolds defined by (i.e. ) which is contained in the intersection of each two classes. The basic classes are distinguished by the following characteristic conditions, respectively
[TABLE]
The class of the Norden manifolds (complex manifolds with Norden metric) is the widest integrable class (i.e. with a vanishing Nijenhuis tensor) and is characterized also by the condition
[TABLE]
Let be the curvature tensor of , i.e.
[TABLE]
Its corresponding (0,4)-tensor with respect to is defined by
and has the following properties
[TABLE]
Any tensor of type (0,4) which satisfies all three conditions in (1.7) is called a curvature-like tensor. Then, the Ricci tensor and the scalar curvature of are obtained by
[TABLE]
A curvature tensor is called a Kähler tensor if . Then, for the corresponding (0,4)-type tensor with respect to , i.e.
we have .
Let be a tensor of type (0,2), and denote by . Consider the following (0,4)-tensors:
[TABLE]
where \mathchoice{\mathbin{\ooalign{\displaystyle\bigcirc\displaystyle\wedge\cr}}{}}{\mathbin{\ooalign{\textstyle\bigcirc\textstyle\wedge\cr}}{}}{\mathbin{\ooalign{\scriptstyle\bigcirc\scriptstyle\wedge\cr}}{}}{\mathbin{\ooalign{\scriptscriptstyle\bigcirc\scriptscriptstyle\wedge\cr}}{}} is the Kulkarni-Nomizu product of two (0,2)-tensors, e.g.
(g\mathchoice{\mathbin{\ooalign{\displaystyle\bigcirc\displaystyle\wedge\cr}}{}}{\mathbin{\ooalign{\textstyle\bigcirc\textstyle\wedge\cr}}{}}{\mathbin{\ooalign{\scriptstyle\bigcirc\scriptstyle\wedge\cr}}{}}{\mathbin{\ooalign{\scriptscriptstyle\bigcirc\scriptscriptstyle\wedge\cr}}{}}S)(X,Y,Z,W)=g(Y,Z)S(X,W)-g(X,Z)S(Y,W)+g(X,W)S(Y,Z)-g(Y,W)S(X,Z). The tensor is curvature-like iff is symmetric, and is curvature-like iff is symmetric and hybrid with respect to , i.e. .
On a pseudo-Riemannian manifold () the Weyl tensor of a curvature-like tensor is given by
[TABLE]
The square norm of is defined by
[TABLE]
An almost Norden manifold is called isotropic Kählerian if .
1.2. Conjugate connections with respect to a metric tensor and statistical manifolds
Let be a pseudo Riemannian manifold, and be an arbitrary linear connection on . Then the linear connection defined by
[TABLE]
is called the conjugate (dual) connection of with respect to . From (1.11) it is easy to see that (. Hence, and are said to be mutually conjugate. Also, from (1.11) it follows that a connection is self-conjugate, i.e. if and only if it is a metric (-compatible) connection, i.e. .
The average connection of two mutually conjugate connections is a metric connection.
Let and be the curvature tensors of and , respectively, and be the average curvature tensor of and , i.e.
[TABLE]
Then, because of the relation , the corresponding (0,4)-type tensor of is curvature-like.
Let be a torsion free (symmetric) connection. Then, it is known that its conjugate connection is also torsion free if and only if the tensor is completely symmetric, i.e.
[TABLE]
Then the same is valid for , i.e. and are both Codazzi pairs. Also, in this case the average connection of and is the Levi-Civita connection of .
The triple is called a statistical manifold if is torsion free and is completely symmetric. Equivalently, a statistical manifold is a pseudo Riemannian manifold equipped with a pair of symmetric conjugate connections. Then, is called a statistical structure on . Hence, a statistical manifolds is a generalization of a pseudo Riemannian manifold.
An almost Norden manifold equipped with a statistical structure will be called a statistical almost Norden manifold.
1.3. Conjugate connections with respect to an almost complex structure
Let be a pseudo Riemannian manifold, and be an almost complex structure on . If is an arbitrary linear connection then the connection defined by
[TABLE]
is called the complex conjugate connection of [1], [6]. From (1.14) it follows that , i.e. and are mutually conjugate relative to .
A connection is self-conjugate with respect to if and only if it is an almost complex connection (-compatible connection), i.e. .
The average connection of two complex conjugate connections is -compatible [1].
By the same manner as in [6], we prove that if is a Norden metric then . Thus, iff .
2. Relations between conjugate connections
on almost Norden manifolds
Let be an almost Norden manifold. In this section, we study relations between the aforementioned types of conjugate connections on .
First, we study the coincidence of conjugate connections with respect the pair of Norden metrics. Let us remark that if and are conjugate with respect to a Norden metric tensor , then by (1.11) it follows that . Hence, in this case iff .
Proposition 2.1**.**
Let and be linear connections on an almost Norden manifold . Then, each two of the following conditions imply the third one:
- (i)
* and are conjugate relative to ;* 2. (ii)
* and are conjugate relative to ;* 3. (iii)
* ().*
Proof.
Let us prove that conditions (i) and (ii) imply (iii). First, we take into account that and are conjugate with respect to and substitute in (1.11). Hence, by covariant differentiation and the definition of , we obtain
[TABLE]
Then, keeping in mind that and are also conjugate with respect to , equality (2.1) implies .
The truthfulness of the other two statements is proved analogously. ∎
Proposition 2.1 yields the following
Corollary 2.1**.**
Let be a statistical almost Norden manifold. Then, is also a statistical almost Norden manifold if and only if ().
Let us remark that if and are simultaneously statistical manifolds, the Levi-Civita connections and of and , respectively, coincide with the average connection of and and hence . The last implies that and are both Kähler Norden manifolds.
Next, we study the coincidence of conjugate connections relative to the metric and the almost complex structure. In this regard, we prove the following
Proposition 2.2**.**
Let be an almost Norden manifold, and be a linear connection on . Then:
- (i)
the conjugate connections of relative to and to coincide if and only if ;
- (ii)
the conjugate connections of relative to and to coincide if and only if .
Proof.
Let us prove (i) (the other statement is proved analogously). The conjugate connections of relative to and to coincide if and only if the connection defined by (1.14) satisfies condition (1.11). Keeping in mind the properties of and , the last condition is equivalent to
[TABLE]
Then, by substituting in (2.2), we obtain , i.e. which completes the proof. ∎
It is well-known that the unique linear connection which is symmetric and metric with respect to a given metric tensor is the Levi-Civita connection induced by this metric tensor. In light of the last fact, Proposition 2.2 yields
Corollary 2.2**.**
Let be an almost Norden manifold, and be a symmetric connection on . Then:
- (i)
the conjugate connections of relative to and coincide if and only if is the Levi-Civita connection of ;
- (ii)
the conjugate connections of relative to and coincide if and only if is the Levi-Civita connection of .
Thus, the conjugate connection of (resp. ) relative to (resp., to ) is its complex conjugate connection.
The case of a completely symmetric connection is considered in the following
Corollary 2.3**.**
Let be an almost Norden manifold, and let and be linear connections on . Then:
- (i)
If is a statistical manifold, and is the conjugate connection of relative to then is a Kähler manifold;
- (ii)
If is a statistical manifold, and is the conjugate connection of relative to then is a Kähler manifold.
Proof.
(i) Since is a statistical manifold, the average connection of and is . But because it is also the average connection of two complex conjugate connections, should be an almost complex connection, i.e. . Hence, is a Kähler manifold.
(ii) By a similar manner, we deduce that is a Kähler Norden manifold, i.e. which implies . Because is symmetric, the last equality yields and hence is also Kählerian. ∎
Based on the results in this section, we conclude that a pair of linear connections and is conjugate with respect to all three structural tensor , and simultaneously iff (which implies ). Linear connections preserving the structural tensors of the manifold by covariant differentiation are called natural(adapted). Hence, is such a connection.
3. Conjugate Connections of the Levi-Civita
Connections induced by the pair of Norden metrics
As seen in the previous section (Corollary 2.2), the Levi-Civita connections induced by the Norden metrics are the unique symmetric linear connections on an almost Norden manifold for which the conjugate connections relative to the associated metric tensor and the almost complex structure coincide. In this section, we study curvature properties of these connections.
Let us consider the conjugate connection of with respect to and , i.e. . We remark that is a metric connection, i.e. .
If by and we denote the corresponding curvature tensors, according to [6], we have . Hence, the average curvature tensor of and defined by (1.12) satisfies the property , meaning that is a Kähler curvature tensor. For (0,4)-type tensors we have
[TABLE]
Next, we focus on the average connection of and which we denote by , i.e. . Since is conjugate to relative to and simultaneously, the average connection satisfies and hence , i.e. is a natural connection. Moreover, it is the well-known Lichnerowicz first canonical connection [13]. In [22], we have obtained the form of the curvature tensor of on an almost Norden manifold as follows
[TABLE]
Then, the last equality and (3.1) yield
Proposition 3.1**.**
On an almost Norden manifold, the average curvature tensor of the conjugate connections and and the curvature tensor of their average connection are related as follows
[TABLE]
In [22], we have shown that ||\nabla^{0}J||^{2}=2g^{il}g^{jk}g\big{(}(\nabla^{0}_{e_{i}}J)e_{k},(\nabla^{0}_{e_{j}}J)e_{l}\big{)} on a manifold in the class of the Norden manifolds. Then, if by and we denote the scalar curvatures of and , respectively, from (1.5) and (3.2), on a Norden manifold we have
[TABLE]
In [21], we have proved that on a manifold in the class the relation is valid. Then, by (1.6) and (3.3) we get
Corollary 3.1**.**
On a Norden manifold belonging to the class () or to is isotropic Kählerian iff .
Analogous results are valid for the Levi-Civita connection of and its conjugate connection relative to and .
Next, using the characteristic condition (1.6) of the class , the form (1.9) of the tensors and , and by straightforward calculations, we obtain
Proposition 3.2**.**
Let be a -manifold. Then, the curvature tensors and of and , respectively, have the form:
[TABLE]
where is the curvature tensor of , and .
We remark that both and are not (0,4)-type curvature-like tensors.
4. Statistical structures on almost Norden manifolds
In this section, we consider statistical structures on almost Norden manifolds by constructing and studying families of completely symmetric linear connections.
Let be a symmetric linear connection, and be its difference tensor with respect to the Levi-Civita connection of , i.e.
[TABLE]
Denote . Then by covariant differentiation we obtain . If is a statistical structure, the last equality and (1.13) imply that the tensor is completely symmetric, i.e. , and . In this case, the connection is said to be completely symmetric.
[TABLE]
Let us remark that in the theory of statistical manifolds the (0,3)-type tensor , which differs from only by a factor, is called the cubic form (skewness tensor) of the manifold.
It is known that equality (4.1) and imply the following relation between the curvature tensors and of and , respectively
[TABLE]
Analogously, (4.2) yields
[TABLE]
where is the curvature tensor of . Then, by (4.3) and (4.4) we obtain [12]
[TABLE]
Also, since is completely symmetric, we have
[TABLE]
Let us denote
[TABLE]
Since satisfies properties (1.7), is a curvature-like tensor.
Taking into account (4.5), (4.6), (4.7) and the form (1.12) of the average curvature tensor (known as the statistical curvature tensor [9]) of and , from (4.3) we verify
Proposition 4.1**.**
On a statistical manifold, the statistical curvature tensor and the curvature tensor are related as follows
[TABLE]
If is flat, then is also flat which imply . Hence, for a flat statistical manifold .
If we consider as the curvature tensor jointly generated by and then in the next statement we give a necessary and sufficient condition for the Weyl tensor to be invariant under the transformation of the Levi-Civita connection into the pair of symmetric conjugate connections .
Corollary 4.1**.**
On a statistical manifold, the Weyl tensors of and coincide iff where is given by (4.7).
Let be an almost Norden manifold, and be a statistical structure on . If we ask for this structure to be compatible with , i.e. (which implies ) we immediately obtain . Hence, almost complex completely symmetric connections exist only on Kähler manifolds. Thus, in order to study wider classes of statistical almost Norden manifolds we will not aim for -compatibility.
4.1. Completely symmetric connections constructed by the metrics and the Lie 1-forms
According to (1.6), an almost Norden manifold which is not in the class has non-vanishing Lie 1-forms and . Thus, on such manifolds, the pairs of Lie 1-forms and Norden metrics can be used to construct difference tensors of completely symmetric linear connections and thus statistical structures. One such family of connections is introduced in the next
Proposition 4.2**.**
On an almost Norden manifold , there exists a four-parametric family of completely symmetric connections defined by (4.1) with difference tensor given by
[TABLE]
* ().*
By (1.9), (4.7), (4.9) and straightforward calculations we obtain
Proposition 4.3**.**
Let be the statistical almost Norden manifold with defined by (4.1) and (4.9). Then, the statistical curvature tensor of the manifold has the form (4.8) where
[TABLE]
and
[TABLE]
Since for the Weyl of it is valid , by Corollary 4.1, equalities (1.9), (1.1) and (4.10) we get the following
Proposition 4.4**.**
Let be the family of linear connections defined by (4.1) and (4.9) with the condition . Then, the Weyl tensors of and coincide.
4.2. Completely symmetric connections constructed by the Lie 1-forms
A family of completely symmetric linear connections with difference tensor depending only on the Lie 1-forms and is presented in the following
Proposition 4.5**.**
On an almost Norden manifold , there exists a four-parametric family of completely symmetric connections defined by (4.1) with difference tensor given by
[TABLE]
* ().*
By (4.7), (4.11) and straightforward calculations we obtain
Proposition 4.6**.**
Let be the statistical almost Norden manifold with defined by (4.1) and (4.11). Then, the statistical curvature tensor of the manifold has the form (4.8) where
[TABLE]
where .
A direct consequence of the last statement and (4.8) is that on manifolds with isotropic Lie vector field with respect to both and , i.e. satisfying , we obtain , and thus the statistical curvature tensor of the statistical structure defined by (4.1) and (4.11) coincides with the curvature tensor of .
Acknowledgement. The author would like to express her gratitude to Professor Dr. C. Udrişte for his suggestion on the topic of this paper.
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