# 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.

### 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 ...

### Famous quotes containing the word manifold:

As one who knows many things, the humanist loves the world precisely because of its manifold nature and the opposing forces in it do not frighten him. Nothing is further from him than the desire to resolve such conflicts ... and this is precisely the mark of the humanist spirit: not to evaluate contrasts as hostility but to seek human unity, that superior unity, for all that appears irreconcilable.
Stefan Zweig (18811942)