Riemannian metric
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| Riemannian metric | |
|---|---|
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| Name | Riemannian metric |
| Field | Differential geometry |
| Introduced | 19th century |
| Notable people | Bernhard Riemann, Elwin Bruno Christoffel, Gregorio Ricci-Curbastro, Tullio Levi-Civita |
Riemannian metric A Riemannian metric is a smooth assignment of an inner product to the tangent space at each point of a smooth manifold, providing notions of length, angle, and volume that underpin modern Differential geometry and inform developments in General relativity, Geometric analysis, Riemannian geometry, and related fields. Originating in the work of Bernhard Riemann and formalized through tools developed by Elwin Bruno Christoffel, Gregorio Ricci-Curbastro, and Tullio Levi-Civita, the concept unifies local linear algebra with global topological and analytical structure on manifolds. Its formal properties connect to curvature tensors, geodesic flows, and variational principles studied by figures such as Henri Poincaré, David Hilbert, and Marston Morse.
Definition
A Riemannian metric on a smooth manifold M is a smooth section g of the bundle of symmetric, positive-definite bilinear forms on tangent spaces TM, assigning to each point p an inner product g_p on T_pM compatible with smooth coordinate changes; this notion was framed in foundational work by Bernhard Riemann and elaborated in tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita. In local coordinates (x^1,...,x^n) the metric is represented by a symmetric matrix (g_{ij}) whose components transform under pullback by diffeomorphisms studied in the context of Sophus Lie's theory and later formalized in the tensor formalism used by Élie Cartan. The metric yields pointwise constructions such as norms, angles, and orthogonality, and global constructions such as length of curves and volume forms invoked by Carl Friedrich Gauss in surface theory and extended by Bernhard Riemann to higher dimensions.
Examples
Canonical examples include the Euclidean metric on ℝ^n used by Isaac Newton's classical mechanics and by Joseph-Louis Lagrange in variational formulations, and the round metric on the sphere S^n central to work by Carl Friedrich Gauss and Bernhard Riemann; these metrics provide explicit g_{ij} matrices with constant coefficients or those induced by embeddings studied by Ferdinand von Lindemann and others. The Minkowski metric of Hermann Minkowski arises as a pseudo-Riemannian analogue in the context of Albert Einstein's formulation of Special relativity and motivates comparison with positive-definite metrics. Other instructive examples include warped product metrics appearing in models by Robert C. Myers and in cosmological spacetimes studied by Georges Lemaître and Alexander Friedmann, and metrics with nontrivial holonomy investigated by Simon Donaldson, Shing-Tung Yau, and Michael Atiyah.
Properties
A Riemannian metric determines a unique Levi-Civita connection characterized by metric-compatibility and torsion-freeness, a fact central to analyses by Tullio Levi-Civita and Elwin Bruno Christoffel, and yields curvature quantities such as the Riemann curvature tensor, Ricci tensor, and scalar curvature used in the proofs and conjectures of James Clerk Maxwell-inspired field theories and modern results by Richard S. Hamilton and Grigori Perelman. The metric induces a distance function producing a complete metric space when the Hopf–Rinow theorem conditions studied by Eberhard Hopf and Wilhelm Rinow are met; completeness properties relate to geodesic extendability explored by Marston Morse and George David Birkhoff. Isometries preserving the metric lead to Lie group actions analyzed by Élie Cartan and Sophus Lie, and rigidity theorems such as those of Alaoglu-style compactness and Mostow-type rigidity were advanced by George Mostow and others.
Levi-Civita connection and curvature
The Levi-Civita connection ∇ associated to a Riemannian metric g is the unique affine connection satisfying ∇g = 0 and torsion(∇) = 0, a characterization proven using Christoffel symbols Γ^k_{ij} introduced by Elwin Bruno Christoffel and applied in work by Tullio Levi-Civita. From ∇ one constructs the Riemann curvature tensor R(X,Y)Z encoding sectional curvatures that generalize Gaussian curvature studied by Carl Friedrich Gauss; contractions yield the Ricci tensor and scalar curvature that play central roles in the Einstein field equations formulated by Albert Einstein and in geometric flows such as Ricci flow developed by Richard S. Hamilton and used by Grigori Perelman to resolve the Poincaré conjecture of Henri Poincaré. Holonomy groups of ∇, classified in Riemannian contexts by results linked to Elie Cartan and later refined by Bertram Kostant and Marcel Berger, govern parallel transport phenomena exploited in constructions by Michael Atiyah and Isadore Singer.
Applications and examples in geometry and physics
In geometry, Riemannian metrics underpin comparison theorems such as those by Shing-Tung Yau and Jeff Cheeger, rigidity results by George Mostow, and moduli problems studied by William Thurston and Grigori Perelman; they provide tools for global analysis connecting to index theorems by Atiyah and Isadore Singer and to minimal surface theory advanced by Joseph Plateau and Jesse Douglas. In physics, metrics model spacetime geometry in Albert Einstein's General relativity where the Einstein tensor built from Ricci curvature determines dynamics in solutions like the Schwarzschild metric analyzed by Karl Schwarzschild and the Friedmann–Lemaître–Robertson–Walker models used by Georges Lemaître and Alexander Friedmann. Riemannian metrics also appear in quantum field theory frameworks examined by Richard Feynman and in string theory contexts developed by Edward Witten and Michael Green, and they inform numerical methods and data analysis in applications by researchers connected to institutions such as Princeton University and Massachusetts Institute of Technology.