On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

1990 ◽  
Vol 39 (3) ◽  
pp. 331-380 ◽  
Author(s):  
Neda Bokan
Keyword(s):  
Riemannian Manifolds ◽  
Complete Decomposition ◽  
Symmetric Connection ◽  
Curvature Tensors

scholarly journals Identities for curvature tensors in generalized Finsler space

Filomat ◽  
2009 ◽  
Vol 23 (2) ◽  
pp. 34-42 ◽  
Author(s):  
Milan Zlatanovic ◽  
Svetislav Mincic
Keyword(s):  
Finsler Space ◽  
Cyclic Symmetry ◽  
Symmetric Connection ◽  
Two Indices ◽  
Curvature Tensors

In the some previous works we have obtained several curvature tensors in the generalized Finsler space GFN (the space with non-symmetric basic tensor and non-symmetric connection in Rund's sence). In this work we study identities for the mentioned tensors (the antisymmetriy with respect of two indices, the cyclic symmetry, the symmetry with respect of pairs of indices).

Pseudo B-symmetric manifolds

2017 ◽  
Vol 14 (09) ◽  
pp. 1750119
Author(s):  
Young Jin Suh ◽  
Carlo Alberto Mantica ◽  
Uday Chand De ◽  
Prajjwal Pal
Keyword(s):  
Riemannian Manifold ◽  
Explicit Form ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Sufficient Condition ◽  
Conformally Flat ◽  
Harmonic Curvature ◽  
Codazzi Type ◽  
Curvature Tensors

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

scholarly journals PSEUDO-RIEMANNIAN JACOBI–VIDEV MANIFOLDS

2007 ◽  
Vol 04 (05) ◽  
pp. 727-738 ◽  
Author(s):  
P. GILKEY ◽  
S. NIKČEVIĆ
Keyword(s):  
Riemannian Manifolds ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Ricci Operator ◽  
Almost Complex ◽  
Algebraic Curvature ◽  
Curvature Tensors

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

Some curvature restricted geometric structures for projective curvature tensors

2018 ◽  
Vol 15 (09) ◽  
pp. 1850157 ◽  
Author(s):  
Absos Ali Shaikh ◽  
Haradhan Kundu
Keyword(s):  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Einstein Manifolds ◽  
Geometric Structures ◽  
Projective Curvature ◽  
Riemannian Case ◽  
Projective Curvature Tensor ◽  
Projective Operator ◽  
Curvature Tensors

The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds. It possesses different geometric properties than other generalized curvature tensors. The main object of the present paper is to study some semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor. The reduced pseudosymmetric type structures for various Walker type conditions are deduced and the existence of Venzi space is ensured. It is shown that the geometric structures formed by imposing projective operator on a (0,4)-tensor is different from that for the corresponding (1,3)-tensor. Characterization of various semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor are obtained on semi-Riemannian manifolds, and it is shown that some of them reduce to Einstein manifolds for the Riemannian case. Finally, to support our theorems, four suitable examples are presented.

scholarly journals On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 150-159 ◽  
Author(s):  
Adara M. Blaga ◽  
Antonella Nannicini
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metallic Structures ◽  
Bisectional Curvature ◽  
Curvature Tensors

AbstractWe study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.

scholarly journals On some properties of non-symmetric connections

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 585-591
Author(s):  
Svetislav Mincic
Keyword(s):  
Covariant Derivative ◽  
Dimensional Manifold ◽  
Symmetric Connection ◽  
Curvature Tensors

On an N-dimensional manifold with non-symmetric connection Lijk four kinds of covariant derivative (1.1) are defined, and four curvature tensors are obtained. In the present paper specially the 3rd and the 4th kind of covariant derivative are studied, particularly their application on ?-symbols.

scholarly journals Geometric Quantities of Manifolds with grassmann structure

2005 ◽  
Vol 180 ◽  
pp. 45-76 ◽  
Author(s):  
N. Bokan ◽  
P. Matzeu ◽  
Z. Rakić
Keyword(s):  
Tensor Product ◽  
Point Of View ◽  
Vector Spaces ◽  
Grassmann Manifolds ◽  
Simple Modules ◽  
Complete Decomposition ◽  
Curvature Tensors ◽  
Torsion Free ◽  
Product Vector

AbstractWe study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds

2018 ◽  
Vol 2020 (23) ◽  
pp. 9559-9592
Author(s):  
Jixiang Fu ◽  
Weimin Sheng ◽  
Lixia Yuan
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Elliptic Partial Differential Equations ◽  
Nonlinear Elliptic ◽  
Noncompact Riemannian Manifolds ◽  
Curvature Tensors ◽  
Curvature Problem ◽  
Negative Sectional Curvature

Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

Sign in / Sign up

Export Citation Format

Share Document