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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...
The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean ...
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. Parallel transport is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a ...
If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N. In differential geometry , pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.
Intuitively, development captures the notion that if x t is a curve in M, then the affine tangent space at x 0 may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve C t in this affine space: the development of x t. In formal terms, let τ 0
An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that J p 2 = −1 at each point p. Only when this local tensor can be patched ...
Any Hilbert space is a Hilbert manifold with a single global chart given by the identity function on . Moreover, since is a vector space, the tangent space to at any point is canonically isomorphic to itself, and so has a natural inner product, the "same" as the one on .