I am trying to prove that the kernel of a push-forward is the fiber.
Let $π : E → M $ be a fiber be bundle with a fiber $F$ . What is the meaning of a tangent space to a bundle? Does it means that if we have a vector, $X$ tangent to curve $\lambda$, that curve must pass to all points of the fiber or in just one point of the bundle?
The definition of the tangent space of the bundle $E$ is the same as the definition of the tangent space to any manifold. A tangent vector at a point $p \in E$ is just an equivalence class of curves $[\alpha]$ with $\alpha(0) = p$. Since $E$ is a fiber bundle, instead of considering all the curves you can consider curves $\alpha$ which start at $p$ (so $\alpha(0) = p$) and stay in the fiber $E_{\pi(p)} = \pi^{-1}(\pi(p))$. Such curves will satisfy $d\pi_{p}([\alpha]) = 0$ and they will span the tangent space $T_{p}(E_{\pi(p)})$ which is the tangent space to the fiber $E_{\pi(p)}$ at $p$.
