Riemannian manifold | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-20

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Riemannian_manifold


00:01:22 1 Introduction
00:02:53 2 Overview
00:06:00 2.1 Riemannian manifolds as metric spaces
00:08:59 2.2 Properties
00:09:23 3 Riemannian metrics
00:13:58 3.1 Examples
00:27:52 3.2 The pullback metric
00:29:42 3.3 Existence of a metric
00:30:00 3.4 Isometries
00:32:11 4 Riemannian manifolds as metric spaces
00:35:22 4.1 Diameter
00:35:48 4.2 Geodesic completeness
00:37:16 5 See also
00:38:18 6 References



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SUMMARY
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In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M, g) is a real, smooth manifold M equipped with an inner product gp on the tangent space TpM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ gp(X|p, Y|p) is a smooth function. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). These terms are named after the German mathematician Bernhard Riemann.  The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.