scholarly journals Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1202 ◽  
Author(s):  
Hülya Aytimur ◽  
Mayuko Kon ◽  
Adela Mihai ◽  
Cihan Özgür ◽  
Kazuhiko Takano
Keyword(s):  
Curvature Tensor ◽  
Natural Condition ◽  
Tensor Field ◽  
Statistical Manifolds ◽  
Chen Inequality

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .

On geometry of Kenmotsu manifolds with N-connection

2019 ◽  
pp. 48-60
Author(s):  
A. Bukusheva
Keyword(s):  
Vector Field ◽  
Coordinate System ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Lie Derivative ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Civita Connection ◽  
Kenmotsu Manifolds ◽  
Levi Civita Connection

A Kenmotsu manifold with a given N-connection is considered. From the integrability of the distribution of a Kenmotsu manifold it follows that the N-connection belongs to the class of the quarter-symmetric connections. Among the N-connections, the class of connections adapted to the structure of the Kenmotsu manifold is specified. In particular, it is proved that an N-connection preserves the structure endomorphism φ of the Kenmotsu manifold if and only if the endomorphisms N and φ commute. A formula expressing the N-connection in terms of the Levi-Civita connection is obtained. The Chrystoffel symbols of the Levi-Civita connection and of the N-connection of the Kenmotsu manifold with respect to the adapted coordinates are computed. The properties of the invariants of the interior geometry of the Kenmotsu manifolds are investigated. The invariants of the interior geometry are the following: the Schouten curvature tensor; the 1-form  defining the distribution D; the Lie derivative 0   L g of the metric tensor g along the vector field ;  the tensor field P with the components given with respect to the adapted coordinate system by the formula Pacd  ncad . The field P is called in the work the Schouten — Wagner tensor. It is proved that the Schouten — Wagner tensor of the interior connection of the Kenmotsu manifold is zero. The conditions that satisfies the endomorphism N defining the metric N-connection are found. At the end of the work, an example of a Kenmotsu manifold with a metric N-connection preserving the structure endomorphism φ is given.

scholarly journals A (CHR)3-flat trans-Sasakian manifold

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Hermitian Manifolds ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

scholarly journals A new curvaturelike tensor field in an almost contact Riemannian manifold II

2018 ◽  
Vol 103 (117) ◽  
pp. 113-128 ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Sasakian Manifold ◽  
Tensor Fields ◽  
Geometrical Properties ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)3-flat Sasakian manifold does not exist.

The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds

2020 ◽  
Vol 27 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Doddabhadrappla G. Prakasha ◽  
Luis M. Fernández ◽  
Kakasab Mirji
Keyword(s):  
Curvature Tensor ◽  
Tensor Field ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Projective Curvature Tensor

AbstractWe consider generalized {(\kappa,\mu)}-paracontact metric manifolds satisfying certain flatness conditions on the {\mathcal{M}}-projective curvature tensor. Specifically, we study ξ-{\mathcal{M}}-projectively flat and {\mathcal{M}}-projectively flat generalized {(\kappa,\mu)}-paracontact metric manifolds and, further, ϕ-{\mathcal{M}}-projectively symmetric generalized {(\kappa\neq-1,\mu)}-paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.

scholarly journals On Infinitesimal Holonomy and Isotropy Groups

1957 ◽  
Vol 11 ◽  
pp. 111-114 ◽  
Author(s):  
Katsumi Nomizu
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Present Note ◽  
General Theorem ◽  
Holonomy Group ◽  
Complete Riemannian Manifold ◽  
Isotropy Group ◽  
Covariant Derivatives ◽  
Derivatives Of

We have proved in [2] that if the restricted homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at every point, then the Riemannian manifold is locally symmetric, that is, the covariant derivatives of the curvature tensor field are zero. The proof of this theorem, however, depended on an insufficiently stated proposition (Theorem 1, [2]). In the present note, we shall give a proof of a more general theorem of the same type.

scholarly journals Metric f-Contact Manifolds Satisfying the (κ, μ)-Nullity Condition

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 891
Author(s):  
Alfonso Carriazo ◽  
Luis M. Fernández ◽  
Eugenia Loiudice
Keyword(s):  
Explicit Expression ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Contact Manifold ◽  
Contact Manifolds

We prove that if the f-sectional curvature at any point of a ( 2 n + s ) -dimensional metric f-contact manifold satisfying the ( κ , μ ) nullity condition with n > 1 is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the ( κ , μ ) nullity condition is of constant f-sectional curvature if and only if μ = κ + 1 and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.

Some inequalities for statistical submanifolds of quaternion Kaehler-like statistical space forms

2019 ◽  
Vol 16 (08) ◽  
pp. 1950129 ◽  
Author(s):  
Mohd. Aquib
Keyword(s):  
Space Forms ◽  
Statistical Manifolds ◽  
Statistical Submanifold ◽  
Totally Real ◽  
Statistical Version ◽  
Chen Inequality ◽  
Statistical Space

Motivated by one of the problems proposed by [Vilcu and Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kaehler-like statistical submersions, Entropy 17 (2015) 6213–6228] in this paper, we study the statistical submanifolds of quaternion Kaehler-like statistical space forms and provide an answer to the problem. Further, we derive the statistical version of Chen inequality for totally real statistical submanifold in such ambient.

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