Ehresmann connection

Remarks

Since $d\pi_e$ is onto with kernel $V_e$, it restricts to an isomorphism $H_e \to T_{\pi(e)}M$. So each tangent vector of $M$ has a unique horizontal lift at each point of the fibre above it. Example: for the product bundle $E = M \times F$ with $\pi$ the first projection, $H_{(p, y)} = T_pM \times \set{0}$ is an Ehresmann connection; $V_{(p,y)} = \set{0} \times T_yF$, and the two are complementary. Non-example: $H = TE$ fails the condition when the fibres have positive dimension, because $H_e \cap V_e = V_e \ne \set{0}$. Equivalently, an Ehresmann connection is a smooth vector-valued $1$-form $\Phi$ on $E$ with $\Phi \circ \Phi = \Phi$ and image $V$; then $H = \ker \Phi$. Horizontal lifts of curves exist locally; if the fibres are not compact they may fail to exist for all times. On a vector bundle, the Ehresmann connections invariant under the scalings of the fibres and compatible with addition correspond to connections in the sense of covariant derivatives.