Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

: The final report covering the research performed under the Air Force Office of Scientific Research by the Dallas relativity group, summarizes information on personnel involved, on the scientific work done, and on the results obtained in the four years of the grant's duration. Topics investigated were in areas of General relativity, cosmology, and electrodynamics.

Exact Solutions of Einstein's Field Equations.

Fundamentals. Hyperplanes. Helly-Type Theorems. Kirchberger-Type Theorems. Special Topics in E2. Families of Convex Sets. Characterizations of Convex Sets. Polytopes. Duality. Optimization. Convex Functions. Solutions, Hints, and References for Exercises. Bibliography. Index.

Convex sets and their applications

Constraints on dark matter density and axion mass from the large-scale structure of spacetime

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray’s decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker.

Perfect-fluid, generalized Robertson-Walker space-times, and Gray’s decomposition

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker.

Perfect-fluid, generalised Robertson-Walker space-times, and Gray's decomposition

The purpose of this paper is to study the W2-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the W2-curvature tensor on a warped product manifold in terms of its relation with W2-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate W2-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a W2-curvature flat warped product manifold are derived. Finally, we study the W2-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are W2-curvature flat.

/pdf/the-w2-curvature-tensor-on-warped-product-manifolds-and-2x6316xz58.pdf

The W2-curvature tensor on warped product manifolds and applications

This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson–Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models.

/pdf/2-killing-vector-fields-on-warped-product-manifolds-41eoutuqja.pdf

2-Killing vector fields on warped product manifolds

This paper deals with the study of perfect fluid spacetimes. It is proven that a perfect fluid spacetime is Ricci recurrent if and only if the velocity vector field of perfect fluid spacetime is parallel and α = β. In addition, in a stiff matter perfect fluid Yang pure space with p + σ ≠ 0, the integral curves generated by the velocity vector field are geodesics. Moreover, it is shown that in a generalized Robertson–Walker perfect fluid spacetime, the Weyl tensor is divergence-free and the gradient of the potential function of the concircular vector field is pointwise collinear with the velocity vector field of perfect fluid spacetime. We also characterize the perfect fluid spacetimes whose Lorentzian metrics are Yamabe and gradient Yamabe solitons, respectively.

Sameh Shenawy

Papers

Perfect-fluid, generalized Robertson-Walker space-times, and Gray’s decomposition

Perfect-fluid, generalised Robertson-Walker space-times, and Gray's decomposition

The W2-curvature tensor on warped product manifolds and applications

2-Killing vector fields on warped product manifolds

Perfect fluid spacetimes and Yamabe solitons