tangent bundle

It follows that the tangent bundle of the 3-sphere is trivial.

Its tangent bundle is called the phase space of the constrained system.

More specifically, a vector field can mean a section of the tangent bundle.

Let M be a manifold with a connection on the tangent bundle.

For instance, the tangent bundle of is a natural bundle.

A similar object is the canonical vector field on the tangent bundle.

For example, is the universal cover of the unit tangent bundle to any hyperbolic surface.

It gives the consistency required to define the tangent bundle in an intrinsic way.

This is equivalent to the tangent bundle being trivial.

T)C defines an ordinary complex structure on the tangent bundle.

The Lagrangian is a function on the tangent bundle.

The geodesic flow defines a family of curves in the tangent bundle.

There is actually a much more elegant way using tangent bundles over M, but we will just stick to this version.

For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.

It may be described also as the dual bundle to the tangent bundle.

The unit tangent bundle carries a variety of differential geometric structures.

In particular, let us consider the tangent bundle of a manifold coordinated by .

Example 1: -i is a Dirac operator on the tangent bundle over a line.

Using V one can characterize the tangent bundle.

The construction may be thought of as defining an analog of the tangent bundle in the following way.

More generally, its generalizations serve as replacements for the (unstable) tangent bundle.

Thus if M has a stably trivial tangent bundle then .

In complex geometry one considers structures on the tangent bundles of manifolds.

The tangent bundle of any Lie group can be trivialized via left (or right) multiplication.

For example, consider the tangent bundle TS for n even.