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Riemannian Manifold

A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x,y) is defined as the length of the shortest curve (geodesic) between x and y.

Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.

See also

Bishop's Inequality, Campbell's Theorem, Cheeger's Finiteness Theorem, Hopf-Rinow Theorem, Pseudo-Riemannian Manifold, Riemannian Metric

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Cite this as:

Weisstein, Eric W. "Riemannian Manifold." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannianManifold.html

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