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# Tangent Bundle

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

 (1)

The tangent bundle is a special case of a vector bundle. As a bundle it has bundle rank , where is the dimension of . A coordinate chart on provides a trivialization for . In the coordinates, ), the vector fields , where , 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 there are two different coordinate charts defined on the same hemisphere, and ,

 (2)
 (3)

with and . The map between the coordinate charts is .

 (4)

The Jacobian of is given by the matrix-valued function

 (5)

which has determinant and so is invertible on .

The tangent vectors transform by the Jacobian. At the point in , a tangent vector corresponds to the tangent vector at in . These two are just different versions of the same element of the tangent bundle.

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