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
Other articles related to "riemannian manifold, riemannian manifolds, manifolds, manifold, riemannian":
... 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 ... All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold ...
... A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion ... The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric ... On a Riemannian manifold one has notions of length, volume, and angle ...
... Pseudo-Riemannian manifold Riemannian bundle metric Riemannian circle Riemannian cobordism Riemannian connection Riemannian connection on a surface ...
... A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map is defined for all, i.e ... an open proper submanifold of any other Riemannian manifold ... The converse is not true, however there exist non-extendable manifolds which are not complete ...
... in some of the most useful examples of Riemannian manifolds ... 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 ...
Famous quotes containing the word manifold:
“Thy love is such I can no way repay,
The heavens reward thee manifold I pray.
Then while we live, in love lets so persever,
That when we live no more, we may live ever.”
—Anne Bradstreet (c. 1612–1672)