A smooth map of manifolds that preserves linear connections.

Question

Let $D$ and $D'$ be two linear connections on manifolds $M$ and $M'$ respectively. If $f:M\to M'$ is a smooth map from $M$ into $M'$, what is meant by "$f$ is a connection preserving" ?.

Thanks in advance.

Answer 1

There is a pullback connection $f^*D'$ on $f^*TM$ given by $(f^*D')_X(f^*Y) = D'_{f_*X}Y$ where $X \in \Gamma(M, TM)$ and $Y \in \Gamma(M', TM')$. If $f$ is such that $f^*TM \cong TM$ (e.g. a local diffeomorphism), then $f^*D'$ and $D$ are two connections on $TM$. In this case, $f$ is said to be connection-preserving if $f^*D' = D$.