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So I was working on a problem and I got stuck. The question is as follows. Let $\pi:E\rightarrow M$ be a vector bundle of rank n with a metric and a metric connection on E. Now suppose that we have a set of curves $\gamma_i:[0,1]\rightarrow M$ such that $\gamma_i(0)=p$ and $\gamma(1)=q$ and let us assume that they converge to a curve $\gamma$.What I want to show is that the associated parallel transports converge.

Here is what I have so far. Let $s_i$ be a sequence of sections over $\gamma_i$ such that $s_i(0)=v$ for all $i$. And so what I did was first selects a local parallel frame $s^i\in\Gamma E$, so that each $s_i$ can be written as

$$s_i=a^i_1(s^1\circ\gamma_i)+...+a^i_n(s^n\circ\gamma_i)$$

$$s=a_1(s^1\circ\gamma)+...+a_n(s^n\circ\gamma)$$

Since these are parallel sections we know that the coefficients are constant. Since the smooth curves converge we know that $s^i\circ\gamma_i$ converges to $s^n\circ\gamma$. What I don't know is how the coefficients are determined relative to the curve so I can use the fact that the curves converge. Any suggestions for what I am not seeing? Also we can assume that the curves are converging in $C^1$ by taking coordinate patches.

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  • $\begingroup$ A "local parallel frame" usually does not exist... $\endgroup$ – Amitai Yuval Oct 10 '15 at 22:10
  • $\begingroup$ So when I say local parallel frame, I am taking the sections on some open set $U\subseteq M$. I am not assuming the frame is global as suggested in my post. I think this exists by solving a system of ODE's locally $\endgroup$ – Enigma Oct 10 '15 at 22:36
  • $\begingroup$ You cannot do that... Think of the tangent bundle of a Riemannian manifold with nonvanishing curvature, for example. Take a point, and take small loops starting and ending at this point. The parallel transport along such a loop is not going to be the identity. Hence, there is no section on a neighborhood, which is parallel. $\endgroup$ – Amitai Yuval Oct 10 '15 at 22:54
  • $\begingroup$ Okay thanks. Then I will try something else then. But if we assume this is a regular local frame, what is the relationship between the coefficients and the smooth curves. That's really the information that I am after. $\endgroup$ – Enigma Oct 10 '15 at 23:07
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We may assume that all paths are contained in a neighborhood on which $E$ is trivial (this statement needs justification. I'm leaving that to you). So, let $b_1,\ldots,b_n$, be a frame. A section $s=s^ib_i$ is parallel along the curve $\gamma$ iff it satisfies the equation$$0=\frac{Ds}{dt}=\frac{ds^i}{dt}b_i+s^i\nabla_{\dot{\gamma}}b_i.$$Hence, parallel transport is determined by taking the unique solution to the above equation with a given initial value. Assuming $C^1$ convergence of the paths $\gamma_i$ guarantees convergence of the corresponding equations, which in turn guarantees convergence of the solutions.

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