## 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 Manifold**s 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

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... Pseudo-

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... A

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**Riemannian Manifold**s

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**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**...

... most useful coordinate charts in some of the most useful examples of

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**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)