**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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