“Tangent space”的意思、由来-开放百科全书

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

Informal description

In differential geometry, one can attach to every point

of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through

. The elements of the tangent space at

are called the tangent vectors at

. This is a generalization of the notion of a bound vector in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself.

For example, if the given manifold is a

-sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport and was used by Dirac.^[1] More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

In algebraic geometry, in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety

that gives a vector space with dimension at least that of

itself. The points

at which the dimension of the tangent space is exactly that of

are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. 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 space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

All the tangent spaces of a manifold may be ‘glued together’ to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.

Formal definitions

The informal description above relies on a manifold's ability to be embedded into an ambient vector space

so that the tangent vectors can ‘stick out’ of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself.^[2]

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition as the velocity of curves

In the embedded-manifold picture, a tangent vector at a point

is thought of as the velocity of a curve passing through the point

. We can therefore define a tangent vector as an equivalence class of curves passing through

while being tangent to each other at

Suppose that

is a

manifold (

) and that

. Pick a coordinate chart

, where

is an open subset of

containing

. Suppose further that two curves

with

are given such that both

are differentiable in the ordinary sense (we call these differentiable curves initialized at ). Then

and

are said to be equivalent at

if and only if the derivatives of

and

coincide. This defines an equivalence relation on the set of all differentiable curves initialized at

, and equivalence classes of such curves are known as tangent vectors of

. The equivalence class of any such curve

is denoted by

. The tangent space of

, denoted by

, is then defined as the set of all tangent vectors at

; it does not depend on the choice of coordinate chart

To define vector-space operations on

, we use a chart

and define a map

. This map turns out to be bijective and may be used to transfer the vector-space operations on

over to

, thus turning the latter set into an

-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart

being used, and in fact it does not.

Definition via derivations

Suppose now that

is a

manifold. A real-valued function

is said to belong to

if and only if for every coordinate chart

, the map

is infinitely differentiable. Note that

is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.

Pick a point

. A derivation at

is defined as a linear map

that satisfies the Leibniz identity

then we obtain a real vector space, which we define as the tangent space

The relation between derivations at a point

and tangent vectors at

is as follows: If

is a differentiable curve initialized at

, then the corresponding derivation

is defined by

(where the derivative is taken in the ordinary sense because

is a function from

Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties. However, instead of examining derivations

from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let

be an algebraic variety with structure sheaf

. Then the Zariski tangent space at a point

is the collection of all

-derivations

, where

is the ground field and

is the stalk of

Definition via cotangent spaces

Again, we start with a

manifold

and a point

. Consider the ideal

that consists of all smooth functions

vanishing at

, i.e.,

. Then

and

are real vector spaces, and

may be defined as the dual space of the quotient space

. This latter quotient space is also known as the cotangent space of

While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry.

is a derivation at

, then

for every

, which means that

gives rise to a linear map

. Conversely, if

is a linear map, then

defines a derivation at

. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.

Properties

is an open subset of

, then

is a

manifold in a natural manner (take coordinate charts to be identity maps on open subsets of

), and the tangent spaces are all naturally identified with

Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector

, one defines the corresponding directional derivative at a point

This map is naturally a derivation at

. Furthermore, every derivation at a point in

is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if

is a tangent vector to

at a point

(thought of as a derivation), then define the directional derivative