Tangent Bundle (original) (raw)

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Every smooth manifold M has a tangent bundle TM, which consists of the tangent space TM_p at all points p in M. Since a tangent space TM_p is the set of all tangent vectors to M at p, the tangent bundle is the collection of all tangent vectors, along with the information of the point to which they are tangent.

TM={(p,v):p in M,v in TM_p} (1)

The tangent bundle is a special case of a vector bundle. As a bundle it has bundle rank n, where n is the dimension of M. A coordinate chart onM provides a trivialization for TM. In the coordinates, (x_1,...,x_n), the vector fields (v_1,...,v_n), where v_i=partial/partialx_i, span the tangent vectors at every point (in the coordinate chart). The transition function from these coordinates to another set of coordinates is given by the Jacobian of the coordinate change.

For example, on the unit sphere, at the point (1,0,0) there are two different coordinate charts defined on the same hemisphere, phi:U_1->S^2 and psi:U_2->S^2,

phi(x_1,x_2)=(cosx_1cosx_2,sinx_1cosx_2,sinx_2) (2)
psi(y_1,y_2)=(sqrt(1-y_1^2-y_2^2),y_1,y_2) (3)

with U_1=(-pi/2,pi/2)×(-pi/2,pi/2) and U_2={(y_1,y_2):y_1^2+y_2^2<1}. The map between the coordinate charts is alpha=psi^(-1) degreesphi.

(y_1,y_2)=alpha(x_1,x_2)=(sinx_1cosx_2,sinx_2) (4)

The Jacobian of alpha:U_1->U_2 is given by the matrix-valued function

[cosx_1cosx_2 -sinx_1sinx_2; 0 cosx_2] (5)

which has determinant cosx_1cos^2x_2 and so is invertible on U_1.

The tangent vectors transform by the Jacobian. At the point (x_1,x_2) in U_1, a tangent vector v corresponds to the tangent vector Jv at alpha(x_1,x_2) in U_2. These two are just different versions of the same element of the tangent bundle.

See also

Calculus, Coordinate Chart, Cotangent Bundle, Directional Derivative, Euclidean Space, Jacobian,Manifold, Tangent Bundle Section, Tangent Space, Tangent Vector, Vector Field, Vector Space Explore this topic in the MathWorld classroom

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Tangent Bundle." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TangentBundle.html

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