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...

mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle...

vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E...

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...

a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at...

connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus...

can also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle, so it is a fiber bundle with structure group GL(n, R). That...

manifold and (TM,πTM,M) its tangent bundle. Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semi-spray on M, if...

(in terms of the tangent bundle, not stable normal bundle) by Whitney. For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse...

O(n)} . The example also works for bundles other than the tangent bundle; if E {\displaystyle E} is any vector bundle of rank k {\displaystyle k} over M...