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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.

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Informal descriptionEdit

 
A pictorial representation of the tangent space of a single point   on a sphere. A vector in this tangent space represents a possible velocity at  . After moving in that direction to a nearby point, one's velocity would then be given by a vector in the tangent space of that nearby point—a different tangent space that is not shown.

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 definitionsEdit

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 curvesEdit

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   at   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   at  . The equivalence class of any such curve   is denoted by  . The tangent space of   at  , denoted by  , is then defined as the set of all tangent vectors at  ; it does not depend on the choice of coordinate chart  .

 
The tangent space   and a tangent vector  , along a curve traveling through  .

To define vector-space operations on  , we use a chart   and define a map   by  . 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 derivationsEdit

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

 

which is modeled on the product rule of calculus.

If we define addition and scalar multiplication on the set of derivations at   by

  •   and
  •  ,

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

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   at   is defined by   (where the derivative is taken in the ordinary sense because   is a function from   to  ).

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   at  .

Definition via cotangent spacesEdit

Again, we start with a   manifold   and a point  . Consider the ideal   of   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   at  .

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.

If   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.

PropertiesEdit

If   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 derivativesEdit

Another way to think about tangent vectors is as directional derivatives. Given a vector   in  , one defines the corresponding directional derivative at a point   by

 

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   in the direction   by

 

If we think of   as the initial velocity of a differentiable curve   initialized at  , i.e.,  , then instead, define   by

 

Basis of the tangent space at a pointEdit

For a   manifold  , if a chart   is given with  , then one can define an ordered basis   of   by

 

Then for every tangent vector  , one has

 

This formula therefore expresses   as a linear combination of the basis tangent vectors   defined by the coordinate chart  .[3]

The derivative of a mapEdit

Every smooth (or differentiable) map   between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces:

 

If the tangent space is defined via differentiable curves, then this map is defined by

 

If, instead, the tangent space is defined via derivations, then this map is defined by

 

The linear map   is called variously the derivative, total derivative, differential, or pushforward of   at  . It is frequently expressed using a variety of other notations:

 

In a sense, the derivative is the best linear approximation to   near  . Note that when  , then the map   coincides with the usual notion of the differential of the function  . In local coordinates the derivative of   is given by the Jacobian.

An important result regarding the derivative map is the following:

Theorem. If   is a local diffeomorphism at   in  , then   is a linear isomorphism. Conversely, if   is an isomorphism, then there is an open neighborhood   of   such that   maps   diffeomorphically onto its image.

This is a generalization of the inverse function theorem to maps between manifolds.

See alsoEdit

ReferencesEdit

  1. ^ Dirac, General Theory of Relativity (1975), Princeton University Press
  2. ^ Chris J. Isham (1 January 2002). Modern Differential Geometry for Physicists. Allied Publishers. pp. 70–72. ISBN 978-81-7764-316-9.
  3. ^ Lerman, Eugene. "An Introduction to Differential Geometry" (PDF). p. 12.

External linksEdit