Pullback Connection is Connection on Smooth Manifold

Theorem

Let $M$ and $\tilde M$ be smooth manifolds with or without boundaries.

Let $\phi : M \to \tilde M$ be a diffeomorphism.

Let $T \tilde M$ and $TM$ be tangent bundles of $\tilde M$ and $M$ respectively.

Let $\tilde \nabla$ be the connection in $T \tilde M$.

Let $\phi^* \tilde \nabla$ be the pullback connection.

Then $\phi^* \tilde \nabla$ is the connection in $TM$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Pullback Connections