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Assignment of vector fields to manifolds
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.
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, the velocity would then be given by a vector in the tangent space of that point--a different tangent space that is not shown.
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 embeddedsubmanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport. Many authors in differential geometry and general relativity use it.  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.
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.
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 via tangent 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 differentiable manifold (with smoothness) 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 where . Again, one needs to check that this construction does not depend on the particular chart and the curve being used, and in fact it does not.
The 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.
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.
For and a differentiable curve such that define (where the derivative is taken in the ordinary sense because is a function from to ). One can ascertain that is a derivation at the point and that equivalent curves yield the same derivation. Thus, for an equivalence class we can define where the curve has been chosen arbitrarily. The map is a vector space isomorphism between the space of the equivalence classes and that of the derivations at the point
Definition via cotangent spaces
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.
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 derivatives
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 point
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 .
The derivative of a map
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: