Theorem: Let $M,N$ be two smooth manifold and $f_1,f_2:M\to N$ are two smooth homotopic maps. If $E$ is a vector bundle on $N$, then $f_1^*E\cong f_2^*E$.

Proof: Let $i_1:M\to M\times[0,1],p\mapsto(p,0)$ and $i_2:M\to M\times[0,1],p\mapsto(p,1)$.

Consider the homotopy $H:M\times[0,1]\to N$ such that $H\circ i_1=f_1,H\circ i_2=f_2$.

So it suffices to proof that for vector bundle $F=H^*E$ on $M\times [0,1]$, $i_1^*F=i_2^*F$.

Given a connection on $E$, and let $\gamma_p:[0,1]\to M\times[0,1],t\mapsto(p,t)$.

Then the parallel transport $P_{0,1,\gamma_p}:F_{(p,0)}\to F_{(p,1)}$ is a linear isomorphism.

Hence it induced a bundle isomorphism $i_1^*F\to i_2^*F,\left(p,v_{(p,0)}\right)\mapsto\left(p,P_{0,1,\gamma_p}\left(v_{(p,0)}\right)\right)$.

Corollary: Vector bundle over contractible smooth manifold is trivial.

Proof: Since the identity map on $M$ is homotopic to the constant map, so $TM\cong M\times T_pM$ is trivial.

In particular, the vector bundle over segment are trivial. This is an very important proposition in Riemannian geometry in my opinion. With this proposition, we can deal with the pull-back tangent bundle on curve as we used to do in multivariable calculus. the connection $\hat{\nabla}_{\frac{\mathrm{d}}{\mathrm{d}t}}$ is the same as the derivative in $\mathbb{R}^n$. And thanks to the compatibility of Levi-civita connection and Riemannian metric, we also reserve the Leibniz rule in $\mathbb{R}^n$. For example, we can write the definition of geodesic and Jacobi field as

$$\gamma^{\prime\prime}=0$$

$$J^{\prime\prime}=-\mathrm{R}(J,\gamma^\prime)\gamma^\prime$$

Moreover, the normal component of $J$ w.r.t. $\gamma^\prime$ is also a Jacobi field since

$$(\langle J,\gamma^\prime\rangle\gamma^\prime)^{\prime\prime}=\langle J^{\prime\prime},\gamma^\prime\rangle\gamma^\prime=-\langle\mathrm{R}(J,\gamma^\prime)\gamma^\prime,\gamma^\prime\rangle\gamma^\prime=-\mathrm{R}(J,\gamma^\prime,\gamma^\prime,\gamma^\prime)\gamma^\prime=0.$$