Riemannian manifold

plural Riemannian manifolds

1. differential geometry; Riemannian geometry; differential geometry; Riemannian geometryA 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.

   3 more examples

   In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded.

   1984, Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, page 55

   Further, a much harder theorem, due to J. Nash [31], says that a separable Riemannian manifold of dimension d can be isometrically embedded in #92;textstyle#92;mathbbRᴺ with N#92;le#123;#92;tfrac 1 2#125;d(d#43;1)(3d#43;11).

   2000, Daniel W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society, page 165

   Before we delve into the general theory of Riemannian manifolds, we pause to give it some substance by introducing a variety of "model Riemannian manifolds" that should help to motivate the general theory.

   2018, John M. Lee, Introduction to Riemannian Manifolds, 2nd edition, Springer, page 55

2. A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product; (more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.
   1. differential geometry; Riemannian geometry; more formallyan ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.

      Synonyms: Riemannian space