Differential geometry MOC

Differential pushforward

Let be a -map. The pushforward is an operation for “pushing forward” data defined on to data defined on . Usually this corresponds to some kind of postcomposition in the sense of Pushforward and pullback of morphisms.

Differential pushforward of a vector field

Let be a vector field on . As a derivation on , the pushforward is defined by #m/def/geo/diff

where is the Differential pullback of a scalar field . This is equivalent to the Tangent map acting on the tangent space or bundle of .

Differential pushforward of a contravariant tensor field

The above may be viewed as a special case of the following. Let be a totally contravariant tensor field. The pushforward is defined by #m/def/geo/diff

for 1-forms , where denotes the differential pullback of a 1-form.

For mixed tensor fields it is in general not possible to define the pushforward, except for the special case of the Differential pullback along a diffeomorphism.

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