Pullback (differential Geometry) - Pullback By Diffeomorphisms

Pullback By Diffeomorphisms

When the map φ between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold. The linear map

can be inverted to give

A general mixed tensor field will then transform using Φ and Φ-1 according to the tensor product decomposition of the tensor bundle into copies of TN and T*N. When M = N, then the pullback and the pushforward describe the transformation properties of a tensor on the manifold M. In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.