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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 space has an interpretation in terms of K[t]/(t 2), the dual numbers for K; in the parlance of schemes, morphisms from Spec K[t]/(t 2) to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x. [3] Therefore, one also talks about tangent vectors. See also: tangent space to a ...
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of is twice the dimension of . Each tangent space of an n-dimensional manifold is an n-dimensional vector space
The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. [2]
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 .
Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor (although the metric tensor can be understood to be a special case of a solder form, as it establishes a mapping between the tangent and cotangent bundles of the base space, i.e. between the horizontal and vertical subspaces of the frame ...
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
Then, for any k-point (), the fiber of : ([] / ()) over p is called the tangent space to F at p. [2] If the functor F preserves fibered products (e.g. if it is a scheme), the tangent space may be given the structure of a vector space over k .