Riemannian geometry

Riemannian geometry
Name	Riemannian geometry
Established	19th century
Founder	Bernhard Riemann

Riemannian geometry is the study of smooth manifolds endowed with a smoothly varying positive-definite metric tensor, integrating ideas from differential calculus, topology, and global analysis to investigate shape, distance, and curvature. Originating in the 19th century, it has deep connections to mathematical physics, global analysis, and several branches of pure mathematics through tools developed by leading figures and institutions.

Definition and basic concepts

Riemannian geometry was founded by Bernhard Riemann and further developed by Elwin Bruno Christoffel, Gregorio Ricci-Curbastro, Tullio Levi-Civita, Élie Cartan, and later contributors such as Hermann Weyl, Shiing-Shen Chern, Marston Morse, John Milnor, Michael Atiyah, Isadore Singer, Shing-Tung Yau, Richard Hamilton, Grigori Perelman, Peter Kazhdan, André Weil, and Anders Hesse. The subject uses the notion of a smooth manifold first formalized by Henri Poincaré and later axiomatized by Oswald Veblen and Hassler Whitney, while analytical foundations draw on work by Karl Weierstrass and David Hilbert. Core objects include tensors introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita, and topological invariants studied by Henri Poincaré and Élie Cartan. The interplay with algebraic structures was shaped by Emmy Noether and André Weil, and modern categorical perspectives involve contributors such as Alexander Grothendieck and Saunders Mac Lane.

Riemannian metrics and manifolds

A Riemannian metric on a smooth manifold modeled in the spirit of Bernhard Riemann is a smooth section of symmetric positive-definite bilinear forms, a concept formalized in contexts influenced by David Hilbert and Elie Cartan. Metrics are studied on examples from Euclid-inspired spaces through curved models like those used by Albert Einstein in General relativity and by Ludwig Boltzmann-inspired statistical manifolds. Important constructions include pullbacks associated with maps studied by Henri Lebesgue-inspired integration theory, and products relevant in the work of John von Neumann and Hermann Weyl. Notions of completeness reference the Hopf–Rinow theorem with antecedents in work by Jaques Hadamard and later refined in global studies by Raoul Bott and Lefshetz-related topologists. Metric compatibility and volume forms connect to the analysis of Bernhard Riemann-type zeta considerations and the spectral theory pioneered by John von Neumann and Mark Kac.

Connections, covariant derivative, and geodesics

The Levi-Civita connection introduced by Tullio Levi-Civita provides the unique torsion-free metric-compatible covariant derivative, building on Christoffel symbols attributed to Elwin Bruno Christoffel. Parallel transport notions align with holonomy studied by Élie Cartan and later by Marcel Berger, whose classification influenced studies by Simon Donaldson and Karen Uhlenbeck. Geodesics minimize energy functionals related to variational methods of Bernard Riemann and Marston Morse, and applications exploit techniques from Srinivasa Ramanujan-era analytic continuation and spectral methods associated with Atle Selberg and Harish-Chandra. Exponential maps and injectivity radii feature in explorations by Georges de Rham and were applied by John Milnor in investigations of exotic spheres and by Mikhail Gromov in systolic inequalities.

Curvature: Riemann, Ricci, and scalar curvature

Curvature tensors originating in Ricci calculus of Gregorio Ricci-Curbastro and Tullio Levi-Civita produce the Riemann curvature tensor, Ricci tensor, and scalar curvature, concepts utilized by Albert Einstein in his field equations and by James Clerk Maxwell-inspired geometrical formulations. Élie Cartan’s moving frames refined sectional curvature ideas influential for Hermann Weyl and André Weil, while comparison tools were expanded by Shiing-Shen Chern and Kunihiko Kodaira. Ricci flow, introduced by Richard Hamilton and used decisively by Grigori Perelman, deforms metrics via Ricci curvature and connects to minimal surface techniques of J. Willard Gibbs-related variational analysis. Scalar curvature obstructions link to index-theoretic results proven by Michael Atiyah and Isadore Singer and more recent developments by Stuart Smale-influenced topologists.

Comparison theorems and global results

Comparison theorems such as those of Rauch, Toponogov, and Bishop–Gromov connect sectional and Ricci curvature bounds to global topology; these theorems were shaped by work from Harald Bohr-era analysts and refined by Mikhail Gromov, Jeff Cheeger, Dennis Sullivan, Yau, and Richard Schoen. Global results include the soul theorem by Jeff Cheeger and James G. Wolf-related contributors, splitting theorems with roots in work by John Milnor and Oswald Veblen, and rigidity results exemplified by Mostow rigidity tied to George Mostow. Techniques draw on comparison geometry, synthetic methods from Alexandre Grothendieck-inspired structuralism, and index-theoretic tools developed by Atiyah and Singer.

Examples and important classes of Riemannian manifolds

Key examples include constant-curvature spaces inspired by Bernhard Riemann and widely studied symmetric spaces classified by Élie Cartan and Marcel Berger, including compact examples related to Élie Cartan’s work and noncompact duals appearing in studies by Harish-Chandra. Kähler manifolds, influenced by Erich Kähler, intertwine with complex geometry as in work by Kunihiko Kodaira and Shing-Tung Yau; Calabi–Yau manifolds central to Edward Witten and Cumrun Vafa arise in string theory contexts linked to Michael Atiyah-type index theorems. Einstein manifolds and solitons appear in results by Richard Hamilton and Grigori Perelman, while spaces with special holonomy were classified by Marcel Berger and further explored by Simon Donaldson and Kronheimer. Homogeneous spaces associated with Lie groups studied by Sophus Lie and Elie Cartan provide fertile examples used in representation theory by Harish-Chandra.

Applications and relations to other fields

Applications span General relativity in Albert Einstein’s theory, gauge theories developed by Chen Ning Yang and Robert Mills, and geometric analysis techniques used in index theory by Michael Atiyah and Isadore Singer. Interactions with topology are central to work by Henri Poincaré and William Thurston, while analytic and probabilistic methods connect to spectral geometry studied by Mark Kac and Peter Sarnak. Mathematical physics applications involve contributions from Edward Witten, Cumrun Vafa, Andrew Strominger, and Juan Maldacena in string theory, whereas global geometric methods inform number-theoretic techniques used by Alexandre Grothendieck-inspired arithmetic geometers and in the Langlands program associated with Robert Langlands. Ongoing research continues across institutions such as Princeton University, Harvard University, Institute for Advanced Study, Clay Mathematics Institute, Courant Institute, and Mathematical Sciences Research Institute.

Category:Differential geometry