pseudo-Riemannian geometry

pseudo-Riemannian geometry
Name	Pseudo-Riemannian geometry
Field	Differential geometry
Introduced	20th century
Notable figures	Bernhard Riemann, Henri Poincaré, Élie Cartan, Albert Einstein, Marcel Grossmann

pseudo-Riemannian geometry is the study of smooth manifolds equipped with a nondegenerate, smooth, symmetric metric tensor of indefinite signature, generalizing Riemannian geometry to contexts where lengths can be negative. It provides the geometric language for many foundational theories such as general relativity, and connects to analytic frameworks developed by figures like Bernhard Riemann, Élie Cartan, and Hermann Weyl. The subject intersects with topology studied by Henri Poincaré, global analysis advanced by Atle Selberg, and mathematical physics influenced by Albert Einstein and Marcel Grossmann.

Definition and Basic Properties

A pseudo-Riemannian manifold is a smooth manifold M endowed with a smooth tensor g of type (0,2) that is symmetric and nondegenerate at every point, with a signature (p,q) specifying p positive and q negative eigenvalues; foundational contributions include work by Bernhard Riemann and formalizations by Elie Cartan and Hermann Weyl. Local existence of coordinate charts follows from the Frobenius theorem and partitions of unity as in constructions by Élie Cartan and tools used by Jean Leray, while global invariants relate to index theorems proven by Michael Atiyah and Isadore Singer. Nondegeneracy implies the metric has an inverse raising and lowering indices, enabling tensor algebra developed in the style of Gregorio Ricci-Curbastro and Tullio Levi-Civita.

Examples and Special Cases

Canonical examples include Minkowski space of signature (1,3) central to special relativity and the Lorentzian metrics on Schwarzschild metric and Kerr metric solutions of Einstein field equations studied by Karl Schwarzschild and Roy Kerr. Riemannian manifolds arise as the positive-definite special case used in works by Bernhard Riemann and Marcel Berger, while ultrahyperbolic metrics appear in analyses by John Wheeler and investigations by Roger Penrose into causal structure. Product manifolds constructed by collaborators of James Clerk Maxwell and models like anti-de Sitter space used in Juan Maldacena's conjecture provide further important examples tied to Conformal field theory and results related to Edward Witten.

Connections and Curvature

Every pseudo-Riemannian manifold admits a unique torsion-free metric connection, the Levi-Civita connection, defined using methods pioneered by Tullio Levi-Civita and leveraged by Élie Cartan; curvature tensors—Riemann, Ricci, and scalar curvature—are central quantities used in proofs by Albert Einstein and index results by Michael Atiyah and Isadore Singer. The decomposition of the curvature by the Weyl tensor follows analyses similar to those of Hermann Weyl, while Cartan's structural equations underpin holonomy classifications studied by M. Berger and later refined by Simon Salamon. Energy conditions in Stephen Hawking and Roger Penrose singularity theorems use Ricci curvature inequalities, and techniques from Richard Hamilton's Ricci flow inform geometric evolution even in indefinite signature contexts explored by Grigori Perelman.

Geodesics and Global Structure

Geodesic equations derive from the Levi-Civita connection and variational principles akin to those used by Joseph-Louis Lagrange and developed in relativity by Albert Einstein and Marcel Grossmann; completeness, conjugate points, and cut loci parallel studies by Marston Morse in Morse theory and influence global results like the Penrose singularity theorem proven by Roger Penrose. Causality theory for Lorentzian manifolds—chronology, causality, and global hyperbolicity—was shaped by work from Stephen Hawking, George Ellis, and Roger Penrose, and relates to splitting theorems analogous to those by Wolfgang Ambrose and Isadore Singer. Topological obstructions to certain signatures involve techniques from Michael Atiyah and Raoul Bott.

Applications in Physics and Mathematics

Pseudo-Riemannian geometry underlies general relativity formulated by Albert Einstein and developed through exact solutions by Karl Schwarzschild, Roy Kerr, and perturbative methods used by Richard Feynman. It appears in modern theoretical frameworks including string theory explored by Edward Witten and Juan Maldacena, and in geometric analysis problems treated by Michael Atiyah and Isadore Singer. Numerical relativity used in gravitational wave predictions credited to collaborations including Kip Thorne and Rainer Weiss relies on pseudo-Riemannian initial value formulations influenced by Yvonne Choquet-Bruhat. In pure mathematics, index theory, global analysis, and holonomy groups studied by Marcel Berger, Simon Salamon, and Karsten Grove extend techniques from pseudo-Riemannian settings to inform results in topology due to Henri Poincaré and Raoul Bott.

Advanced Topics and Generalizations

Topics include holonomy classification in indefinite signature developed following work by Marcel Berger and later authors, spin geometry and Dirac operators on pseudo-Riemannian manifolds building on Paul Dirac and Friedrich Hirzebruch's themes, and analysis of hyperbolic differential operators informed by Lars Hörmander and Jean Leray. Generalizations include pseudo-Finsler and pseudo-Hermitian structures influenced by studies of Elie Cartan and developments in complex geometry by Kunihiko Kodaira and Shing-Tung Yau, and connections to noncommutative geometry advanced by Alain Connes. Current research engages with problems posed by Edward Witten and conjectures related to geometric flows and singularity formation addressed by Grigori Perelman and Richard Hamilton.

Category:Differential geometry