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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 ...
In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle.
: ′ will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold ′, which carries the inner product from the surface with negative curvature. φ {\displaystyle \varphi } will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,
A Kähler manifold is a Riemannian manifold of even dimension whose holonomy group is contained in the unitary group (). [3] Equivalently, there is a complex structure on the tangent space of at each point (that is, a real linear map from to itself with =) such that preserves the metric (meaning that (,) = (,)) and is preserved by parallel transport.
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 ...
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic. If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT space. Consequently, its fundamental group Γ = π 1 (M) is Gromov hyperbolic. This has many implications for the ...
Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. [1] The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point.