Tangent bundle

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of a differentiable manifold

The vector bundle , also denoted , whose total space is given by the union of the tangent spaces to at , consisting of the tangent vectors to , and with projection mapping to the point . A section of the tangent bundle is a vector field on (cf. Vector field on a manifold). An atlas on the manifold is defined through an atlas of the manifold . The bundle is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.

Associated with the tangent bundle is the frame bundle of the manifold , which is a principal bundle. The bundle dual to the tangent bundle , called the cotangent bundle, consists of the cotangent spaces to . Its sections are the differential or Pfaffian forms (cf. Pfaffian form).

A differentiable mapping induces a morphism of tangent bundles ; the corresponding mapping of the total spaces is called the tangent mapping to (or differential of ). In particular, if is an immersion (cf. Immersion of a manifold), then is a subbundle of the induced vector bundle . The quotient bundle is called the normal bundle of the immersion. Dually, if is a submersion, then the quotient bundle is called a subbundle of . If one chooses for and , and respectively, and , then is called the tangent bundle of second order.

If is trivial, then is called a parallelizable manifold.


[1] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)


The tangent mapping (also called differential) induced by a differentiable mapping is given by

, , , where is the algebra of smooth functions on and a tangent vector is seen as a special kind of -linear mapping .

In terms of local coordinates and the " /xi-notation" (cf. Tangent vector), the matrix of is given by the Jacobian matrix of the expression for in the local coordinates.

There are many notations in use for the differential . Some common ones are: , , , , . The last one, in case is a function , "sort of agrees" in both notation and name with as the differential -form on defined by (cf. Differential; Differential form). Using the " /xi and dxi" notation (cf. Tangent vector), the differential -form is given in local coordinates by

(where is the result of applying the tangent vector to ). Let be the coordinate for . Then is given by

because .

The differential -frame bundle over has as fibre over the set of all -frames in . (An -frame at is a set of independent vectors in . The frame bundle is the differential -frame bundle. A frame on is a section of the frame bundle and a framed manifold is a manifold provided with a frame.)


[a1] M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965)
[a2] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3
[a3] F. Brickell, R.S. Clark, "Differentiable manifolds" , v. Nostrand-Reinhold (1970)
[a4] L. Auslander, R.E. MacKenzie, "Introduction to differentiable manifolds" , Dover, reprint (1977)
[a5] R. Hermann, "Geometry, physics, and systems" , M. Dekker (1973)
[a6] Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian)
How to Cite This Entry:
Tangent bundle. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article