Canonical connections with an algebraic curvature tensor field on naturally reductive spaces

1992 ◽  
Vol 43 (3) ◽  
Author(s):  
AnnaMaria Pastore
Keyword(s):  
Curvature Tensor ◽  
Tensor Field ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature ◽  
Naturally Reductive

scholarly journals GEOMETRIC REALIZATIONS OF KAEHLER AND OF PARA-KAEHLER CURVATURE MODELS

2010 ◽  
Vol 07 (03) ◽  
pp. 505-515 ◽  
Author(s):  
M. BROZOS-VÁZQUEZ ◽  
P. GILKEY ◽  
E. MERINO
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Constant Scalar Curvature ◽  
Kaehler Manifold ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature.

scholarly journals The nonexistence of rank4IP tensors in signature(1,3)

2002 ◽  
Vol 31 (5) ◽  
pp. 259-269
Author(s):  
Kelly Jeanne Pearson ◽  
Tan Zhang
Keyword(s):  
Vector Space ◽  
Curvature Tensor ◽  
Bilinear Form ◽  
Symmetric Operator ◽  
Symmetric Bilinear Form ◽  
Real Vector Space ◽  
Real Vector ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature

LetVbe a real vector space of dimension4with a nondegenerate symmetric bilinear form of signature(1,3). We show that there exists no algebraic curvature tensorRonVso that its associated skew-symmetric operatorR(⋅)has rank4and constant eigenvalues on the Grassmannian of nondegenerate2-planes inV.

scholarly journals On Lorentzian spaces of constant sectional curvature

2018 ◽  
Vol 103 (117) ◽  
pp. 7-15
Author(s):  
Vladica Andrejic
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Jacobi Operator ◽  
Constant Sectional Curvature ◽  
Duality Principle ◽  
Lorentzian Manifolds ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature ◽  
Curvature Tensors

We investigate Osserman-like conditions for Lorentzian curvature tensors that imply constant sectional curvature. It is known that Osserman (moreover zwei-stein) Lorentzian manifolds have constant sectional curvature. We prove that some generalizations of the Rakic duality principle (Lorentzian totally Jacobi-dual or four-dimensional Lorentzian Jacobi-dual) imply constant sectional curvature. Moreover, any four-dimensional Jacobi-dual algebraic curvature tensor such that the Jacobi operator for some nonnull vector is diagonalizable, is Osserman. Additionally, any Lorentzian algebraic curvature tensor such that the reduced Jacobi operator for all nonnull vectors has a single eigenvalue has a constant sectional curvature.

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 On quasi-Clifford Osserman curvature tensors

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1241-1247
Author(s):  
Vladica Andrejic ◽  
Katarina Lukic
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Duality Principle ◽  
Algebraic Curvature ◽  
Curvature Tensors

We consider pseudo-Riemannian generalizations of Osserman, Clifford, and the duality principle properties for algebraic curvature tensors and investigate relations between them. We introduce quasi- Clifford curvature tensors using a generalized Clifford family and show that they are Osserman. This allows us to discover an Osserman curvature tensor that does not satisfy the duality principle. We give some necessary and some sufficient conditions for the total duality principle.

scholarly journals A DECOMPOSITION OF THE CURVATURE TENSOR FIELD ON THE ABBENA-THURSTON MANIFOLD

2014 ◽  
Vol 36 (4) ◽  
pp. 795-803
Author(s):  
Joon-Sik Park
Keyword(s):  
Curvature Tensor ◽  
Tensor Field

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.

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