scholarly journals Affine connections on singular warped products

2021 ◽  
pp. 2150076
Author(s):  
Yong Wang
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Product Structure ◽  
Warped Products ◽  
Affine Connections ◽  
Almost Product Structure

In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated with the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated with the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in [O. Stoica, The geometry of warped product singularities, Int. J. Geom. Methods Mod. Phys. 14(2) (2017) 1750024, arXiv:1105.3404 .] to singular multiply warped products.

scholarly journals The geometry of warped product singularities

2017 ◽  
Vol 14 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Ovidiu Cristinel Stoica
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Big Bang ◽  
Warping Function ◽  
Warped Products ◽  
Riemann Curvature ◽  
The Big Bang

In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

The Local Möbius Equation and Decomposition Theorems in Riemannian Geometry

2002 ◽  
Vol 45 (3) ◽  
pp. 378-387
Author(s):  
Manuel Fernández-López ◽  
Eduardo García-Río ◽  
Demir N. Kupeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Warped Product ◽  
Product Structure ◽  
Twisted Product ◽  
Partial Differential ◽  
Decomposition Theorems

AbstractA partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

scholarly journals Geometric Inequalities for Warped Products in Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 923
Author(s):  
Bang-Yen Chen ◽  
Adara M. Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Point Of View ◽  
Warping Function ◽  
Research Subject ◽  
Warped Products ◽  
Geometric Inequalities ◽  
Comprehensive Survey ◽  
Optimal Inequality ◽  
Active Research

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2 . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.

scholarly journals Convolution of Riemannian manifolds and its applications

2002 ◽  
Vol 66 (2) ◽  
pp. 177-191 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Warped Products ◽  
Basic Properties ◽  
Segre Embedding

It is well-known that warped products play some important roles in differential geometry as well as in physics. In this article we extend the notion of warped product to the notion of convolution of Riemannian manifolds. We study the basic properties of convolutions of Riemannian manifolds. We also apply the notion of convolution to establish and characterise the Euclidean version of Segre embedding.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

Pointwise pseudo-slant submanifolds of a Kenmotsu manifold

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5833-5853 ◽  
Author(s):  
Viqar Khan ◽  
Mohammad Shuaib
Keyword(s):  
Present Article ◽  
Warped Product ◽  
Shape Operator ◽  
Warped Products ◽  
Canonical Structure ◽  
Slant Submanifold ◽  
Kenmotsu Manifold ◽  
Slant Submanifolds ◽  
Kenmotsu Manifolds

In the present article, we have investigated pointwise pseudo-slant submanifolds of Kenmotsu manifolds and have sought conditions under which these submanifolds are warped products. To this end first, it is shown that these submanifolds can not be expressed as non-trivial doubly warped product submanifolds. However, as there exist non-trivial (single) warped product submanifolds of a Kenmotsu manifold, we have worked out characterizations in terms of a canonical structure T and the shape operator under which a pointwise pseudo slant submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

scholarly journals RETRACTED Warped Product Pointwise Semi-slant Submanifolds of Sasakian Manifol Retracted

2021 ◽  
Vol 45 (5) ◽  
pp. 721-738
Author(s):  
ION MIHAI ◽  
◽  
SIRAJ UDDIN ◽  
АДЕЛА MIHAI
Keyword(s):  
Warped Product ◽  
Warped Products ◽  
Sasakian Manifolds ◽  
Slant Submanifolds ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds of Sasakian manifolds.

Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold

2020 ◽  
Vol 70 (1) ◽  
pp. 151-160
Author(s):  
Amalendu Ghosh
Keyword(s):  
Scalar Curvature ◽  
Warped Product ◽  
Constant Scalar Curvature ◽  
Product Structure ◽  
Kenmotsu Manifold ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Yamabe Soliton

AbstractIn this paper, we study Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. First, we prove that if a Kenmotsu metric is a Yamabe soliton, then it has constant scalar curvature. Examples has been provided on a larger class of almost Kenmotsu manifolds, known as β-Kenmotsu manifold. Next, we study quasi Yamabe soliton on a complete Kenmotsu manifold M and proved that it has warped product structure with constant scalar curvature in a region Σ where ∣Df∣ ≠ 0.

Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows

1989 ◽  
Vol 9 (3) ◽  
pp. 427-432 ◽  
Author(s):  
Renato Feres ◽  
Anatoly Katok
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Tensor Fields ◽  
Invariant Tensor ◽  
Affine Connections

AbstractWe consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

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