Riemannian manifolds

Noun

Riemannian manifold (plural Riemannian manifolds)

1. (differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;
   • By definition, a Riemannian manifold M has at each point p a tangent space T#95;pM equipped with a positive-definite inner product, g#95;p; information about these inner products is encoded in the Riemannian metric tensor, g.
2. (differential geometry, Riemannian geometry, more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.

   Synonyms:

   Riemannian space