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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 ...
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 ...
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]
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.
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R n. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of ...
Real or complex finite-dimensional linear, affine and projective spaces are also smooth manifolds. At each one of its points, a smooth path in a smooth manifold has a tangent vector that belongs to the manifold's tangent space at this point. Tangent spaces to an n-dimensional smooth manifold are n-dimensional linear spaces. The differential of ...
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .
The term framed manifold (occasionally rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle. A related notion is the concept of a π-manifold. [4]