What is 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.

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Some articles on riemannian manifold:

Riemannian Manifolds As Metric Spaces - Geodesic Completeness
... A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map is defined for all, i.e ... isometric to an open proper submanifold of any other Riemannian manifold ... The converse is not true, however there exist non-extendable manifolds which are not complete ...
List Of Things Named After Bernhard Riemann - Riemannian
... Pseudo-Riemannian manifold Riemannian bundle metric Riemannian circle Riemannian cobordism Riemannian connection Riemannian connection on a surface Riemannian cubic ...
Differentiable Manifold - Structures On Manifolds - (Pseudo-)Riemannian Manifolds
... A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion ... the form of a symmetric 2-tensor called the Riemannian metric ... On a Riemannian manifold one has notions of length, volume, and angle ...
Manifolds With Additional Structure - Riemannian Manifolds
... Main article Riemannian manifolds To measure distances and angles on manifolds, the manifold must be Riemannian ... A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product ⟨⋅,⋅⟩ in a manner which varies smoothly from point to point ... All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold ...
List Of Coordinate Charts
... most useful coordinate charts in some of the most useful examples of Riemannian manifolds ... notion 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 that we have defined a ...

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