Let $X$ be a manifold and $\pi:E\rightarrow X$ a vector bundle over $X$ equipped with a metric $\left\langle \cdot,\cdot\right\rangle $.
Let $f:[0,1]\rightarrow M$ be a smooth map, and consider the pullback bundle $f^{*}E\rightarrow[0,1]$. This is the vector bundle whose fiber over $t\in[0,1]$ is
$ (f^{*}E)_{t}=\left\{ (t,v)\,:\, v\in E_{f(t)}\right\} . $
Let $\phi:f^{*}E\rightarrow E$ denote the map $\phi(t,v)=v$.
Let $\nabla$ denote a connection on $E$, and let $D:=f^{*}\nabla$ denote the induced connection on $f^{*}E$.
$D$ is defined as follows: suppose $F\in\Gamma(f^{*}E)$ is a section of $f^{*}E$, so that $\phi(F(t))\in E_{f(t)}$ for all $t\in[0,1]$. Fix $s\in[0,1]$, and let $v$ denote any vector field on $X$ with $v(f(s))=\phi(F(s))$.
Then by definition
$ (D_{\partial_{t}}F)(s)=\left(s,(\nabla_{\dot{f}(s)}v)(f(s))\right)\mbox{ as elements of }(f^{*}E)_{s}, $
the points being that the right-hand side of the above expression is independent of the choice of $v$.
Assume now that $f(0)=f(1)$. Then if $F\in\Gamma(f^{*}E)$ then $\phi(F(1))$ and $\phi(F(0))$ both lie in the same vector space $E_{f(0)}$. My question is: does the following generalization of the fundamental theorem of calculus always hold?
$ |\phi(F(1))-\phi(F(0))| \le \int_{0}^{1}|\phi(D_{\partial_{t}}F)(t)|dt. $