Meaning of Riemannian manifold | Babel Free

Definitions

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

Examples

“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).”

“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.”

CEFR level