Riemannian manifold

Riemannian manifold
Name	Riemannian manifold
Founder	Bernhard Riemann
Field	Differential geometry

Riemannian manifold A Riemannian manifold is a smooth manifold endowed with an inner product on each tangent space varying smoothly, introduced by Bernhard Riemann in the 19th century and developed in the work of Elie Cartan, Hermann Weyl, and Élie Joseph Cartan. It provides the geometric framework underlying the theories of Carl Friedrich Gauss, Gregorio Ricci-Curbastro, Tullio Levi-Civita, and Marston Morse, and it plays a central role in the mathematical foundations of Albert Einstein's general relativity, the geometric analysis of Shing-Tung Yau, and the index theory of Michael Atiyah and Isadore Singer.

Definition and basic concepts

A Riemannian manifold is a smooth manifold M equipped with a Riemannian metric g that assigns to each point p in M an inner product g_p on the tangent space T_pM varying smoothly in local charts, a notion rooted in the coordinate methods of Bernhard Riemann and refined in the tensor formalism of Gregorio Ricci-Curbastro and Tullio Levi-Civita. The metric enables measurement of lengths of tangent vectors, angles, and volumes, linking to the work of Carl Friedrich Gauss on surfaces, Élie Cartan's method of moving frames, and Hermann Weyl's contributions to invariant theory. Fundamental associated objects include the Levi-Civita connection ∇ (unique torsion-free metric connection), the Riemann curvature tensor R, the Ricci tensor Ric, and the scalar curvature S, concepts advanced by Marcel Berger, Shiing-Shen Chern, and James Simons.

Examples and constructions

Classical examples arise from constant-curvature spaces: the Euclidean space R^n, the sphere S^n with the round metric studied by Gauss and Riemann, and the hyperbolic space H^n relevant to the work of Nikolai Lobachevsky and Henri Poincaré. Other constructions include product manifolds studied in the context of John Milnor's work, warped products appearing in the models of Einstein and Roy Kerr, and quotient manifolds such as compact hyperbolic manifolds examined by William Thurston and Grigori Perelman. Important families include symmetric spaces classified by Élie Cartan and Armand Borel, Kähler manifolds central to Eugenio Calabi and Shing-Tung Yau, and homogeneous spaces linked to Sophus Lie and Élie Cartan's theory of transformation groups.

Riemannian metrics and tensor fields

A Riemannian metric g is a smooth (0,2)-tensor field that is positive-definite at each point, a formulation developed in the tensor language of Gregorio Ricci-Curbastro and Tullio Levi-Civita and extended in the global analysis of John Nash and Michael Freedman. Operations on tensor fields use the Levi-Civita connection, leading to covariant derivatives ∇ and parallel transport studied by Élie Cartan and Marcel Berger; curvature tensors R, Ric, and S encode geometric invariants instrumental to theorems of Richard Hamilton and Grigori Perelman on Ricci flow and Thurston's geometrization conjecture. Volume forms and Hodge theory, developed by W. V. D. Hodge and Hermann Weyl, interact with metrics to produce analytic tools such as the Laplace–Beltrami operator, central in the spectral geometry of Mark Kac and Peter Buser.

Geodesics and curvature

Geodesics are curves locally minimizing length and are characterized by the geodesic equation defined via the Levi-Civita connection, with foundational results by Carl Friedrich Gauss, Bernhard Riemann, and Elie Cartan; Jacobi fields describe variations of geodesics and were studied by Carl Gustav Jacobi. Curvature notions—sectional curvature, Ricci curvature, and scalar curvature—originate in the curvature tensor and were pivotal in the proofs of the theorems of Bonnet-Myers, Cartan-Hadamard, and Synge, and in the sphere theorem contributions of Marcel Berger and Karsten Grove. Comparison techniques using Jacobi field estimates underlie the Bishop-Gromov volume comparison by Richard Bishop and Mikhail Gromov and the Toponogov triangle comparison theorem associated with Victor Toponogov.

Comparison theorems and global results

Global Riemannian geometry synthesizes local curvature information into topological and metric consequences, exemplified by the Hopf-Rinow theorem, the Cheeger-Gromoll splitting theorem by Jeff Cheeger and Detlef Gromoll, and the Cheeger finiteness theorem by Jeff Cheeger. Comparison theorems such as Rauch's comparison theorem, Bishop-Gromov volume comparison, and Myers' theorem connect to rigidity results like Mostow rigidity, Torsten Carleman's techniques, and Perelman's work on Ricci flow resolving Thurston's geometrization conjecture and the Poincaré conjecture. Index theorems by Michael Atiyah and Isadore Singer relate curvature to topology via elliptic operators, with extensions in the work of Henri Cartan, Jean-Pierre Serre, and Raoul Bott.

Applications and related structures

Riemannian manifolds are applied across mathematics and physics: in general relativity formulated by Albert Einstein, in gauge theory influenced by Michael Atiyah and Simon Donaldson, and in string theory developments of Edward Witten and Cumrun Vafa; they underpin techniques in global analysis by Lars Hörmander and spectral geometry by Mark Kac. Related structures include pseudo-Riemannian manifolds central to Hermann Minkowski and Einstein, Kähler and Calabi–Yau manifolds relevant to Eugenio Calabi and Shing-Tung Yau, symplectic manifolds tied to André-Marie Ampère (historical antecedents) and Jean-Marie Souriau, and metric measure spaces studied by Yann Ollivier and John Lott in synthetic geometry. Modern research connects Riemannian geometry to the calculus of variations of Leonhard Euler, probability on manifolds in the work of Kiyosi Itô, and geometric group theory advanced by Mikhail Gromov.

Category:Differential geometry