Tangent space

In differential geometry, a tangent space is a vector space, which approximates a differentiable manifold at point linearly. Let be a differentiable curve, then:

A tangent vector. The tangent vectors at a point span a vector space, the tangent space. See also tangent bundle.

In algebraic geometry, one has to modify this definition approach to account for singular points and changing dimensions.

This article deals only with the tangent space over a differentiable manifold in the sense of differential geometry.

• 3.1 tangent vectors as directional derivatives
• 3.2 The total derivative of a mapping

Survey

The simplest is a differentiable manifold to illustrate that (for example, s) embedded as sub- manifold in a Euclidean space. As an example, the sphere ( = sphere ) to serve in the. The tangent space at a point is then a plane that divides exactly one point with the ball and at this point is a tangent plane to the sphere.

A vector field assigns to each point of a manifold a vector of the associated tangent space. For example, you could specify a vector field, the wind speed and direction on the earth's surface.

All tangent space of a manifold can be summarized as tangent bundle of; the tangent bundle is itself a manifold; its dimension is twice as large as that of.

Formal definitions

In the literature it is common to specify the same three different definitions corresponding to a geometric, algebraic and a theoretical physics ( hinarbeitenden on tensors ) perspective. However, the vivid geometric access proves in the application as it than to handle the most tedious.

The next two on the geometric definition of algebraic definitions of the tangent space, however, only work for manifolds of class, but not with.

Geometric Definition: Direction fields of curves

Given a -dimensional manifold with a point from an open neighborhood of and a map.

It has a differentiable curve in such a differentiable curve is. The derivative exists so. This derivation is a vector in. Curves for the matches, form an equivalence class. Such an equivalence class is called a tangent vector of in and writes it. The tangent space is the set of all such tangent vectors; It can be shown that it does not depend on the choice of the map.

It remains to show that becomes a vector space by way of declaration of vector addition and scalar multiplication. Therefore one is by the figure, the function on the right-hand side is an arbitrary representative of the equivalence class. We now show that this map is bijective and transfers with their help, the vector space operations by; it also shows that this construction of the election of the board is independent.

First Algebraic Definition: generalized derivatives

Be a manifold. A function belongs to the class if is infinitely differentiable for each card. The thus defined is an associative algebra.

We fix a point in. A Derivation of a linear map, which has for all and in the ( product rule for analog) following property: . These derivations naturally form a real vector space; this is the tangent space.

The relationship between the previously defined tangent vectors and the derivative ions is as follows: if a curve with tangent vector, then the corresponding derivation is ( with the derivation in the usual sense, as a function of after is ).

Second Algebraic Definition: dual space of I/I2

Be again a manifold and a point in. Consider now the ideal of which consists of all even functions that map to. Then and real vector spaces, and is defined as the dual space of the quotient space. referred to is also called cotangent space ( see below).

While this definition is the most abstract, it is also the one that can be transferred to other situations, the easiest example, varieties, as they are considered in algebraic geometry.

Be a derivation of. Then, for each in ( because there are with, thus ), which induces a linear map. Conversely, a derivation, if a linear map. This shows that the match on derivations and the tangent space over defined.

Tangent space in algebraic geometry

The two algebraic definitions work just as well for algebraic varieties, in which case the tangent space is also called the Zariski tangent space. Unlike manifolds algebraic varieties may also have singularities, where the tangent has then a higher dimension than in smooth points.

Properties

If an open subset of, it can be regarded as a manifold in a natural way. All cards here are the identity, and the tangent space can be identified with the.

Tangent vectors as directional derivatives

One view of tangent is to see them as directional derivatives. For a vector in one defines the directional derivative of a smooth function by a point

This figure is obviously a derivation. In fact, even every Derivation of () of this form. So there is a bijection between vectors ( intended as a tangent vector at the point ) and the derivative ions.

Since tangent vectors can be defined in a general manifold as derivations, it is only natural to see them as directional derivatives. Concretely, it is possible to define the directional derivative in the direction of element for following a tangent of a point ( as seen Derivation ):

Do we see in terms of the geometric definition of the tangent space as a curve, we write

The total derivative of a mapping

Every differentiable map between two differentiable manifolds induces a linear map

Between the corresponding tangent spaces, defined by

For the geometric definition of the tangent space and

For by the definition of derivative ions.

In a sense, the total dissipation is the best linear approximation of f in a neighborhood of p. In local coordinates one can represent the total derivative as a Jacobian matrix.

If the tangent map is surjective, so the Jacobian has full rank everywhere, so you call the underlying function submersion; is the tangent map is injective immersion.

An important result concerning Tangentialabbildungen is the set:

This is a generalization of the theorem on inverse functions on maps between manifolds.

Cotangent space

Since the tangent space at the point of the manifold carries the structure of a vector space, one can form the dual space from him. This space is called the cotangent space and usually notated. Following the recent definition of the space must therefore be isomorphic. The cotangent space also plays a very important role in differential geometry. So you can, for example, the total differential

Understood as a linear mapping which assigns to each tangent vector the directional derivative in its direction. The total differential is thus an element of Kotangentialraums of the point.