Covariant derivative

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold equipped with an affine connexion $\nabla$ . The covariant derivative gives a coördiante-independent notion of the directional derivative of any tensor field along a vector field. #m/def/geo/diff

We extend the connexion as a differential operator $\nabla : 𝔛 (𝑀) \to T 11 (𝑀)$ to tensor fields of arbitrary rank to get an $ℝ$ -linear map

T 𝑝 𝑞 (𝑀) \to T 𝑝 𝑞 + 1 (𝑀) 𝑋 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \mapsto \nabla 𝑐 𝑋 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞

which is compatible with

for $𝑓 \in 𝐶 𝛼 (𝑀)$ ;

index contraction so that $\nabla 𝑑 (𝐴 𝑎 1 \dots 𝑐 \dots 𝑎 𝑝 𝑏 1 \dots 𝑐 \dots 𝑏 𝑞) = \nabla 𝑑 𝐴 𝑎 1 \dots 𝑐 \dots 𝑎 𝑝 𝑏 1 \dots 𝑐 \dots 𝑏 𝑞$ for $𝐴 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \in T 𝑝 𝑞 (𝑀)$ ;
the Leibniz rule so that $\nabla 𝑒 (𝐴 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 𝐵 𝑐 1 \dots 𝑐 𝑝' 𝑑 1 \dots 𝑑 𝑞') = (\nabla 𝑒 𝐴 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞) 𝐵 𝑐 1 \dots 𝑐 𝑝' 𝑑 1 \dots 𝑑 𝑞' + 𝐴 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 (\nabla 𝑒 𝐵 𝑐 1 \dots 𝑐 𝑝' 𝑑 1 \dots 𝑑 𝑞')$ for $𝐴 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \in T 𝑝 𝑞 (𝑀)$ and $𝐵 𝑎 1 \dots 𝑎 𝑝' 𝑏 1 \dots 𝑏 𝑞' \in T 𝑝' 𝑞' (𝑀)$ .

Bibliography

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