In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... A differentiable manifold is a generalization of Euclidean space to extend the meaning of differentiabillity. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

An element of T(M) is a pair (x,v) where x ∈ M and v ∈ T_x(M), the tangent space at x. There is a natural projection The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

which sends (x,v) to the base point x.

Topology and smooth structure

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 it its own right. The dimension of T(M) is twice the dimension of M. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. ... A differential structure, also known as a smooth structure, describes important properties of a manifold which lie in the realm between topology and geometry. ...

Each tangent space of an n-dimensional vector space is an n-dimensional vector space. As a set then, T(M) is isomorphic to M × Rⁿ. As a manifold, however, T(M) is not always diffeomorphic to the product manifold M × Rⁿ. When this happens the tangent bundle is said to be trivial. Just as manifolds are locally modelled on Euclidean space, tangent bundles are locally modelled on M × Rⁿ. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

If M is an n-dimensional manifold, then it comes equipped with an atlas of charts (U_α, φ_α) where U_α is an open set in M and In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

is a homeomorphism. These local coordinates on U give rise to an isomorphism between T_xM and Rⁿ for each x ∈ U. We may then define a map In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

by

We use these maps to define the topology and smooth structure on T(M). A subset A of T(M) is open iff is open in R²ⁿ for each α. These maps are then homeomorphisms between open subsets of T(M) and R²ⁿ and therefore serve as charts for the smooth structure on T(M). 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 R²ⁿ. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

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 n-dimensional manifold M may be defined as a rank n vector bundle over M whose transition functions are given by the Jacobian of the associated coordinate transformations. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

Examples

The simplest example is that of Rⁿ. In this case the tangent bundle is trivial and isomorphic to R²ⁿ. Another simple example is the unit circle, S¹. The tangent bundle is of the circle is also trivial and isomorphic to S¹ × R. Geometrically, this is a cylinder of infinite height. Illustration of a unit circle. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ...

Unfortunately, the only tangent bundles that can be readily visualized are those of the real line R and the unit circle S¹, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence not easily visualizable.

Perhaps the simplest example of a nontrivial tangent bundle is that of the unit sphere S². That the tangent bundle of S² is nontrivial is a consequence of the hairy ball theorem. The hairy ball theorem of algebraic topology states that, in laymans terms, one cannot comb the hair on a ball in a smooth manner. This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing...

Vector fields

A smooth assignment of a vector at each point of a manifold is called a vector field. Specifically, a vector field on a manifold M is a smooth map Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

such that the image of x, denoted V_x, lies in T_x(M), the tangent space to x. In the language of fiber bundles, such a map is called a section. A vector field on M is therefore a section of the tangent bundle of M. In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...

The set of all vector fields on M is denoted by Γ(TM). Vector fields can be added together pointwise

(V + W)_x = V_x + W_x

and multiplied by smooth functions on M

(fV)_x = f(x)V_x

to get other vector fields. The set of all vector fields Γ(TM) then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted C^∞(M). In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...

A local vector field on M is a local section of the tangent bundle. That is, a local vector field is defined only on some open set U in M and assigns to each point of U a vector in the associated tangent space. The set of local vector fields on M forms a structure known as a sheaf of real vector spaces on M. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

See also

• pushforward
• vector field
• vector bundle
• cotangent bundle
• frame bundle

In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ...

External links

• MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle

References

• John M. Lee, Introduction to Smooth Manifolds, (2003) Springer-Verlag, New York. ISBN 0-387-95495-3.
• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3540426272
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X