In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,
where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at .
There is a natural projection
defined by . This projection maps each element of the tangent space to the single point .
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of . By definition, a manifold is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold is framed if and only if the tangent bundle is stably trivial, meaning that for some trivial bundle the Whitney sum is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if is a smooth function, with and smooth manifolds, its derivative is a smooth function .
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. If is an open contractible subset of , then there is a diffeomorphism which restricts to a linear isomorphism from each tangent space to . As a manifold, however, is not always diffeomorphic to the product manifold . When it is of the form , then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean space, tangent bundles are locally modeled on , where is an open subset of Euclidean space.
If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts , where is an open set in and
is a diffeomorphism. These local coordinates on give rise to an isomorphism for all . We may then define a map
We use these maps to define the topology and smooth structure on . A subset of is open if and only if
is open in for each These maps are homeomorphisms between open subsets of and and therefore serve as charts for the smooth structure on . The transition functions on chart overlaps are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of .
The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an -dimensional manifold may be defined as a rank vector bundle over whose transition functions are given by the Jacobian of the associated coordinate transformations.
The simplest example is that of . In this case the tangent bundle is trivial: each is canonically isomorphic to via the map which subtracts , giving a diffeomorphism .
Another simple example is the unit circle, (see picture above). The tangent bundle of the circle is also trivial and isomorphic to . Geometrically, this is a cylinder of infinite height.
The only tangent bundles that can be readily visualized are those of the real line and the unit circle , both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.
A simple example of a nontrivial tangent bundle is that of the unit sphere : this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.
A smooth assignment of a tangent vector to each point of a manifold is called a vector field . Specifically, a vector field on a manifold is a smooth map
such that with for every . In the language of fiber bundles, such a map is called a section . A vector field on is therefore a section of the tangent bundle of .
The set of all vector fields on is denoted by . Vector fields can be added together pointwise
and multiplied by smooth functions on M
to get other vector fields. The set of all vector fields then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted .
A local vector field on is a local section of the tangent bundle. That is, a local vector field is defined only on some open set and assigns to each point of a vector in the associated tangent space. The set of local vector fields on forms a structure known as a sheaf of real vector spaces on .
The above construction applies equally well to the cotangent bundle – the differential 1-forms on are precisely the sections of the cotangent bundle , that associate to each point a 1-covector , which map tangent vectors to real numbers: . Equivalently, a differential 1-form maps a smooth vector field to a smooth function .
Since the tangent bundle is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:
In general, the th order tangent bundle can be defined recursively as .
A smooth map has an induced derivative, for which the tangent bundle is the appropriate domain and range . Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives .
A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.
On every tangent bundle , considered as a manifold itself, one can define a canonical vector field as the diagonal map on the tangent space at each point. This is possible because the tangent space of a vector space W is naturally a product, since the vector space itself is flat, and thus has a natural diagonal map given by under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold is curved, each tangent space at a point , , is flat, so the tangent bundle manifold is locally a product of a curved and a flat Thus the tangent bundle of the tangent bundle is locally (using for "choice of coordinates" and for "natural identification"):
and the map is the projection onto the first coordinates:
Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.
If are local coordinates for , the vector field has the expression
More concisely, – the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on , not on , as only the tangent directions can be naturally identified.
Alternatively, consider the scalar multiplication function:
The derivative of this function with respect to the variable at time is a function , which is an alternative description of the canonical vector field.
The existence of such a vector field on is analogous to the canonical one-form on the cotangent bundle. Sometimes is also called the Liouville vector field, or radial vector field. Using one can characterize the tangent bundle. Essentially, can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.
There are various ways to lift objects on into objects on . For example, if is a curve in , then (the tangent of ) is a curve in . In contrast, without further assumptions on (say, a Riemannian metric), there is no similar lift into the cotangent bundle.
The vertical lift of a function is the function defined by , where is the canonical projection.
In differential geometry, one can attach to every point of a smooth manifold, , a vector space called the cotangent space at . Typically, the cotangent space, is defined as the dual space of the tangent space at , , although there are more direct definitions. The elements of the cotangent space are called cotangent vectors or tangent covectors.
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.
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
Suppose that φ : M → N is a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback, and is frequently denoted by φ∗. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using φ.
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ, , at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at φ(x), . Hence it can be used to push tangent vectors on Mforward to tangent vectors on N. The differential of a map φ is also called, by various authors, the derivative or total derivative of φ.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding.
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms.
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold, given the structure of a complex vector space via the almost complex structure of the complex manifold .
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.
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