Differential pullback along a diffeomorphism

Let $𝜑 : 𝑀 \to 𝑁$ be a $𝐶 𝛼$ -diffeomorphism. For a mixed tensor field $𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \in T 𝑝 𝑞 (𝑁)$ , the pullback $𝜑 * 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \in T 𝑝 𝑞 (𝑀)$ is defined by #m/def/geo/diff

(𝜑 * 𝑇) 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 (𝜔 1) 𝑎 1 \dots (𝜔 𝑝) 𝑎 𝑝 (𝑣 1) 𝑏 1 \dots (𝑣 𝑞) 𝑏 𝑞 = 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 (𝜑 * 𝜔 1) 𝑎 1 \dots ((𝜑 - 1) * 𝑣 𝑞) 𝑏 𝑞 .

The pushforward can be defined via the inverse so that

(𝜑 - 1) * = 𝜑 * : T 𝑝 𝑞 (𝑁) \to T 𝑝 𝑞 (𝑀) .

Further terminology

If $𝜑 \in A u t 𝖬 𝖺 𝗇 𝛼 (𝑀)$ and $𝜑 * 𝑇 = 𝑇$ , we say $𝜑$ is a symmetry transformation of the tensor field $𝑇$ .
In the particular case $𝜑 * 𝑔 = 𝑔$ for the metric tensor, we say $𝜑$ is an isometry of $𝑀$ .

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