Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold

Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ...

M in which each tangent space is equipped with an inner product

Inner product space

In mathematics [i], an inner product space is a vector space [i] with additional structure, an inner ...

g in a manner which varies smoothly from point to point. This allows one to define various notions such as angle

Angle

An angle is the figure formed by two rays [i] sharing a common endpoint [i], called the vertex [i] ...

s, lengths of curves, area

Area

Area is a physical quantity [i] expressing the size of a part of a surface [i]. ...

s, curvature

Curvature

Curvature refers to a number of loosely related concepts in different areas of geometry....

, gradient

Gradient

A generalization of these concepts is the gradient in vector calculus [i]; and this article is mostly ab ...

s of functions and divergence of vector field

Vector field

In mathematics [i] a vector field is a construction in vector calculus [i] which associates a vector [i] ...

s.

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In Riemannian geometry, a Riemannian manifold is a real differentiable manifold