Let $M, N$ be two differentiable manifolds such that M is a fiber bundle over $N$ with projection $\pi : M \to N$. Let $U \subset M$ be an open set and $\Lambda \subset U$. For $x \in \Lambda$ consider the tangent space $T_x M$. Is there a connection between $T_xM$ and $T_{\pi(x)} N$?
If I suppose that $T_x M = E^-_x \bigoplus E^+_x$, can I say that there is such a decomposition for $T_{\pi(x)} N$, i. e. $T_{\pi(x)} N = \tilde{E}^{-}_{\pi(x)} \bigoplus \tilde{E}^{+}_{\pi(x)}$?
I don't understand the point of introducing either $U$ or $\Lambda$ here. What is true in general is that if $F \to M \xrightarrow{\pi} N$ is a smooth fiber bundle then it induces a short exact sequence
$$0 \to V \to T(M) \to \pi^{\ast}(T(N)) \to 0$$
of vector bundles on $M$, where $V$ is the vertical tangent bundle.