This is a follow up to my question about Ehresmann connections.
Let $F\to E\stackrel p \to B$ be a fiber bundle. Then $V:=\ker dp \subset TE$ is a subbundle. Furthermore, $i^*TE = V$. A connection is nothing else than defining a splitting $TE = H \oplus V$.
Is it true that independently of the connection, there always is an isomorphism $p^* TB \to H$? If yes, how is it defined?