Semi-Riemannian Geometry With Applications to Relativity
This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.
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CHAPTER 11 HOMOGENEOUS AND SYMMETRIC SPACES
CHAPTER 12 GENERAL RELATIVITY COSMOLOGY
CHAPTER 13 SCHWARZSCHILD GEOMETRY
CHAPTER 14 CAUSALITY IN LORENTZ MANIFOLDS
FUNDAMENTAL GROUPS AND COVERING MANIFOLDS
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algebra apply assertion assume basis called causal closed compact complete components condition consider constant contained converges coordinate system Corollary corresponding covering curvature curve defined Definition derivative diffeomorphism differential direct element energy equation equivalent example Exercise exists expression fact Figure fixed follows formula function geodesic geometry given gives grad hence holds hypersurface identity implies initial integral isometry isomorphism Killing Lemma lift linear Lorentz manifold matrix meet metric tensor natural neighborhood Newtonian normal null geodesic null vectors observer open set operator orbit orientation parallel particle particular plane positive preceding preserves projection Proof properties Proposition Prove relative result Riemannian scalar product segment semi-Riemannian manifold sequence simply connected smooth space spacelike spacetime speed starting subgroup submanifold suffices suppose symmetric tangent vector theorem timelike unique unit usual variation vector field zero