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Riemannian Manifold
Amanifold possessing ametric tensor. For a complete Riemannian manifold, themetric
is defined as the length of the shortest curve (geodesic) between
and
.
Every complete Riemannian manifold isboundedly compact. This is part of or a consequence of theHopf-Rinow theorem.
See also
Bishop's Inequality,Campbell's Theorem,Cheeger's Finiteness Theorem,Hopf-Rinow Theorem,Pseudo-Riemannian Manifold,Riemannian MetricExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Riemannian Manifold."FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/RiemannianManifold.html