Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M {displaystyle M} is a manifold T M {displaystyle TM} which assembles all the tangent vectors in M {displaystyle M} . As a set, it is given by the disjoint union of the tangent spaces of M {displaystyle M} . That is, where T x M {displaystyle T_{x}M} denotes the tangent space to M {displaystyle M} at the point x {displaystyle x} . So, an element of T M {displaystyle TM} can be thought of as a pair ( x , v ) {displaystyle (x,v)} , where x {displaystyle x} is a point in M {displaystyle M} and v {displaystyle v} is a tangent vector to M {displaystyle M} at x {displaystyle x} . There is a natural projection defined by π ( x , v ) = x {displaystyle pi (x,v)=x} . This projection maps each tangent space T x M {displaystyle T_{x}M} to the single point x {displaystyle x} . 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 (a fiber bundle whose fibers are vector spaces). A section of T M {displaystyle TM} is a vector field on M {displaystyle M} , and the dual bundle to T M {displaystyle TM} is the cotangent bundle, which is the disjoint union of the cotangent spaces of M {displaystyle M} . By definition, a manifold M {displaystyle M} is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum T M ⊕ E {displaystyle TMoplus E} 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 f : M → N {displaystyle f:M ightarrow N} is a smooth function, with M {displaystyle M} and N {displaystyle N} smooth manifolds, its derivative is a smooth function D f : T M → T N {displaystyle Df:TM ightarrow TN} . 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 T M {displaystyle TM} is twice the dimension of M {displaystyle M} . Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If U {displaystyle U} is an open contractible subset of M {displaystyle M} , then there is a diffeomorphism T U → U × R n {displaystyle TU ightarrow U imes mathbb {R} ^{n}} which restricts to a linear isomorphism from each tangent space T x U {displaystyle T_{x}U} to { x } × R n {displaystyle {x} imes mathbb {R} ^{n}} . As a manifold, however, T M {displaystyle TM} is not always diffeomorphic to the product manifold M × R n {displaystyle M imes mathbb {R} ^{n}} . When it is of the form M × R n {displaystyle M imes mathbb {R} ^{n}} , 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 U × R n {displaystyle U imes mathbb {R} ^{n}} , where U {displaystyle U} is an open subset of Euclidean space. If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts ( U α , ϕ α ) {displaystyle (U_{alpha },phi _{alpha })} where U α {displaystyle U_{alpha }} is an open set in M {displaystyle M} and