Differential geometry MOC

Differential pullback

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

Differential pullback of a scalar field

Let be a scalar field on . The pullback is defined by #m/def/geo/diff

for , i.e. .

Differential pullback of a covariant tensor field

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

for vector fields , where denotes the Differential pushforward of a vector field.

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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