For a path $\gamma:[0,1]\to B$, a section $s(t)$ of $\gamma^*E$ is parallel when
$$
\dot s+A(\dot\gamma)s=0.
$$
Existence and uniqueness for this linear ordinary differential equation define the <parallel transport>
$$
\mathcal P_\gamma^{\mathcal A}:E_{\gamma(0)}\longrightarrow E_{\gamma(1)}.
$$
Transport along the reversed path solves the inverse initial-value problem, so
$$
\mathcal P_{\bar\gamma}^{\mathcal A}
=(\mathcal P_\gamma^{\mathcal A})^{-1}.
$$

If $P(t)$ denotes transport from $0$ to $t$, then
$$
\mu(t)=P(t)\mu(0)P(t)^{-1}
$$
satisfies the horizontal equation for the induced endomorphism connection. Uniqueness therefore gives
$$
\mathcal P_\gamma^{\operatorname{End}(\mathcal A)}(\mu)
=\mathcal P_\gamma^{\mathcal A}\,
\mu\,
(\mathcal P_\gamma^{\mathcal A})^{-1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-115/3/c/solution