Riemannian manifold

Etymology

Definitions

1. 1

   A real, smooth differentiable manifold whose each point has a tangent space equipped with…

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

The neighborhood