Tangent bundle
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.
References
[1] | C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) |
Comments
The tangent mapping (also called differential) induced by a differentiable mapping
is given by
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,
,
, 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.)
References
[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) |
Tangent bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_bundle&oldid=41514