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.

Equivalents

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