Differential pushforward

Let $𝜑 : 𝑀 \to 𝑁$ be a $𝐶 𝛼$ -map. The pushforward $𝜑 *$ is an operation for “pushing forward” data defined on $𝑀$ to data defined on $𝑁$ . Usually this corresponds to some kind of postcomposition in the sense of Pushforward and pullback of morphisms.

Differential pushforward of a vector field

Let $𝑣 \in 𝔛 (𝑀)$ be a vector field on $𝑀$ . As a derivation on $𝐶 𝛼 (𝑁)$ , the pushforward $𝜑 * 𝑣 \in 𝔛 (𝑁)$ is defined by #m/def/geo/diff

(𝜑 * 𝑣) (𝑓) = 𝑣 (𝜑 * 𝑓)

where $𝜑 * 𝑓 = 𝑓 \circ 𝜑$ is the Differential pullback of a scalar field $𝑓 \in 𝐶 𝛼 (𝑀)$ . This is equivalent to the Tangent map acting on the tangent space or bundle of $𝑀$ .

Differential pushforward of a contravariant tensor field

The above may be viewed as a special case of the following. Let $𝑇 \in T 𝑝 0 (𝑀)$ be a totally contravariant tensor field. The pushforward $𝜑 * 𝑇 \in T 𝑝 0 (𝑁)$ is defined by #m/def/geo/diff

(𝜑 * 𝑇) 𝑎 1 \dots 𝑎 𝑝 (𝜔 1) 𝑎 1 \dots (𝜔 𝑝) 𝑎 𝑝 : = 𝑇 𝑎 1 \dots 𝑎 𝑝 (𝜑 * 𝜔 1) 𝑎 1 \dots (𝜑 * 𝜔 𝑝) 𝑎 𝑝

for 1-forms $(𝜔 1) 𝑎, \dots, (𝜔 𝑝) 𝑎 \in Ω 1 (𝑀)$ , where $𝜑 *$ 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.

Tangent map

#state/tidy | #lang/en | #SemBr

Differential pushforward

Differential pushforward of a vector field

Differential pushforward of a contravariant tensor field

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