Differential pullback

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

Differential pullback of a scalar field

Let $𝑓 \in 𝐶 𝛼 (𝑁)$ be a scalar field on $𝑁$ . The pullback $𝜑 * 𝑓 \in 𝐶 𝛼 (𝑀)$ is defined by #m/def/geo/diff

(𝜑 * 𝑓) (𝑝) : = 𝑓 (𝜑 (𝑝))

for $𝑝 \in 𝑀$ , i.e. $𝜑 * 𝑓 = 𝑓 \circ 𝜑$ .

Differential pullback of a covariant tensor field

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

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

for vector fields $(𝑣 1) 𝑎, \dots, (𝑣 𝑝) 𝑎 \in 𝔛 (𝑀)$ , where $𝜑 *$ 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.

Tangent map

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Differential pullback

Differential pullback of a scalar field

Differential pullback of a covariant tensor field

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