Horizontal lifts of $\gamma$ and $\gamma^{-1}$?

Question

Let $p:E\longrightarrow B$ be a surjective submersion and $\sigma: p^*(TB)\longrightarrow TE$ be a complete connection. For a curve $\gamma:[a, b]\longrightarrow B$ and a point $e\in E_{\gamma(a)}$ we the horizontal lift of $\gamma$ at $e$ which is a horizontal curve $\gamma_e^h: [a, b]\longrightarrow E$ such that $p\circ \gamma_h^e=\gamma$ and $\gamma_e^h(0)=e$.

What is the relationship between the horizontal lifts of $\gamma$ and $\gamma^{-1}$? Here $\gamma^{-1}(t):=\gamma(1-t)$.

I thought it should be something like $$(\gamma^{-1})^h_e=\gamma^h_{(\gamma^{-1})^h_e(1)},$$ but it does not seem to work.

Thanks.

Answer 1

The horizontal lift of $\gamma^{-1}$ at $\gamma^{-1}(0)=\gamma^h(1)$ is ${{\gamma^h}}^{-1}(t)=\gamma^h(1-t)$. This curve is tangent to the distribution which defines the connection and verifies $p({{\gamma^h}}^{-1}(t))=\gamma^{-1}(t)$.