Definition:Pullback Connection

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

Let $T \tilde M$ be the tangent bundle of $\tilde M$.

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

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

Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields on $M$.

Let $\phi^* \tilde \nabla : \map {\mathfrak{X} } M \times \map {\mathfrak{X} } M \to \map {\mathfrak{X} } M$ be the mapping such that:


 * $\forall X, Y \in \map {\mathfrak{X} } M : \paren {\phi^* \tilde \nabla}_X Y := \map {\paren {\phi^{-1} }_* } {\map {\tilde \nabla_{\phi_* X} } {\phi_* Y} }$

where $\phi_*$ and $\phi^*$ stand for the pushforward and pullback respectively.

Then $\phi^* \tilde \nabla$ is called the pullback connection.