A Class of Riemannian Manifolds with Special Form of Curvature Tensor (original) (raw)

On some classes of Riemannian manifolds

arXiv: Differential Geometry, 2019

We study several classes of Riemannian manifolds which are defined by imposing a certain condition on the Ricci tensor. We consider the following cases: Ricci recurrent, Cotton, quasi Einstein and pseudo Ricci symmetric condition. Such conditions can be interpreted as overdetermined PDE systems whose unknowns are the components of the Riemannian metric, and perhaps in addition some auxiliary functions. Hence even if the dimension of the manifold is small it is not easy to compute interesting examples by hand, and indeed very few examples appear in the literature. We will present large families of nontrivial examples of such manifolds. The relevant PDE systems are first transformed to an involutive form. After that in many cases one can actually solve the resulting system explicitly. However, the involutive form itself already gives a lot of information about the possible solutions to the given problem. We will also discuss some relationships between the relevant classes.

A New Class of Golden Riemannian Manifold

International electronic journal of geometry, 2020

In this paper, we introduce a new class of almost Golden Riemannian structures and study their essential examples as well as their fundamental properties. Next, we investigate a particular type belonging to this class and we establish some basic results for Riemannian curvature tensor and the sectional curvature. Concrete examples are given.

Some Curvature Properties of -Manifolds

Abstract and Applied Analysis, 2013

The object of the present paper is to study -manifolds with vanishing quasi-conformal curvature tensor. -manifolds satisfying Ricci-symmetric condition are also characterized.

Riemannian manifolds with Einstein-like metrics

1985

1 n this thesis, we investigate propaties of manifolds with Riemannian metrics which satisfy conditions more general than those of J:'instein metrics, including the Lauer as special cases. Fhe /:"instein condition is weU known for being the l:"uln-l.agrange equation of a vw iational problem. '/'here is twt a great deal of difference between such metrics and melrics with Ricci tensor parallel for the Latter are locaUy Riemannian products of the former. More general classes of metrics considered include Ricci-Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into thr•ee parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner /Kal] while the second and pan of the third is ktwwn to Derdzinski / Del,2]. Next we construct the metrics mentioned above on spheres of odd dimension.'J'he construction is similar to Jensen's /Jell but more direct and is due essentiaUy to Gray and Vanhecke /GV}. In this way we obtain ,beside the standard metric, the second l:.'instein metric of Jensen. As for the Ricci-Codazzi metrics, they are essentiaUy Jiinstein, but the Ricci cyclic parallel mell ics seem to form a larger class. FinaUy,we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute aU the corresponding metrics on the quotient spaces associated with G2.

Jawarneh Manifold with Semi Symmetric Metric Connection

Jawarneh manifold along with semi-symmetric metric connectionhave been studied, and some of its geometric properties are derived, and some application in general relativity are mentioned.Also a new example of Jawarneh manifold is given.

Metric Ricci Curvature for Manifolds

Geometry, 2013

We introduce a metric notion of Ricci curvature for P L manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.

Investigation on some classes of warped product manifolds

2017

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2). Mathematics Subject Classification (2010). 53C15, 53C21, 53C25, 53C35.