Search results
Results from the WOW.Com Content Network
The singular points of are those where the "test to be a manifold" fails. See Zariski tangent space. Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent ...
Given an n-dimensional smooth manifold M, and a point p ∈ M, a contact element of M with contact point p is an (n − 1)-dimensional linear subspace of the tangent space to M at p. [2] [3] A contact element can be given by the kernel of a linear function on the tangent space to M at p.
One is often interested only in C p-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Man p (A). Similarly, the category of C p-manifolds modeled on a fixed space E is denoted Man p (E). One may also speak of the category of smooth manifolds, Man ∞, or the category of analytic manifolds, Man ω.
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
The vector fields λ(A), λ(B), λ(C) form a basis of the tangent space at each point of G. Similarly the left invariant vector fields ρ(A), ρ(B), ρ(C) form a basis of the tangent space at each point of G. Let α, β, γ be the corresponding dual basis of left invariant 1-forms on G. [51]
The Euclidean space with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to Standard n {\displaystyle n} -dimensional hyperbolic space H n {\displaystyle \mathbb {H} ^{n}} is a Cartan–Hadamard manifold with constant sectional curvature equal to − 1. {\displaystyle -1.}
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 .
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.