Riemannian Manifold

Riemannian Manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product on each tangent space that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then is a smooth function. The family of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds comprises the subject called Riemannian geometry.

A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

Read more about Riemannian Manifold:  Introduction, Overview, Riemannian Metrics, Riemannian Manifolds As Metric Spaces

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... in some of the most useful examples of Riemannian manifolds ... of a coordinate chart is fundamental to various notions of a manifold which are used in mathematics ... In order of increasing level of structure topological manifold smooth manifold Riemannian manifold and semi-Riemannian manifold For our purposes, the key feature of the last two examples is ...

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