A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself....

associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Let E → X be a real vector bundle of rank k over a topological...

to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by...

2n. The tangent bundle is where tangent vectors lie, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle. One can...

vector bundle in question is the tangent bundle T M {\displaystyle TM} , the unit sphere bundle is known as the unit tangent bundle. A sphere bundle is partially...

setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields...

the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M)...

geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle of a smooth manifold....

point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than...

inverse Jacobian. A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space...