Open AccessBook
Semi-Riemannian Geometry With Applications to Relativity
TLDR
In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.Abstract:
Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.read more
Citations
More filters
Journal ArticleDOI
The Geometry of Algorithms with Orthogonality Constraints
TL;DR: The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms and developers of new algorithms and perturbation theories will benefit from the theory.
Book ChapterDOI
Semi-Riemannian Geometry
TL;DR: In this paper, the basics of differentiable manifolds and semi-Riemannian geometry for the applications in general relativity are developed. But the applicability of these manifolds to general relativity is not discussed.
Journal ArticleDOI
Stationary Black Holes: Uniqueness and Beyond
TL;DR: Developments in the subject are reviewed and the uniqueness theorem for the Einstein-Maxwell system is discussed in light of the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories.
Journal ArticleDOI
On Smooth Cauchy Hypersurfaces and Geroch's Splitting Theorem
Antonio N. Bernal,Miguel Sánchez +1 more
TL;DR: In this article, the existence of a smooth spacelike Cauchy hypersurface S and a global diffeomorphism between M and R × S was shown, where S is a smooth space.
Journal ArticleDOI
Gravitational Lensing from a Spacetime Perspective
TL;DR: In this paper, the theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations, where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature, and the basic equations and relevant techniques for calculating the position, shape, and brightness of images in an arbitrary general-relativistic spacetime.