Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 3

For $\pi :E\to B$, the vertical space at $p\in E$ is $V_{p}E=\ker (d\pi {)}_p$. A horizontal subspace is a complementary subspaceA connection is a smooth choice of such complements, linear with respect to the vector-bundle structure. Equivalently, it is a smooth horizontal distribution as in horizontal subspace of a vector bundle connection.

Put $b=\pi (p)$ and $u=(d\pi {)}_{p}v$. Choose a vector field $X$ near $b$ with $X(b)=u$, multiply it by a bump function, and extend it by zero to $B$. At each $q\in E$, the restrictionis an isomorphism. Define $V(q)$ as the unique horizontal lift of $X(\pi (q))$. Smoothness of the horizontal distribution makes $V$ smooth, and uniqueness gives $V(p)=v$.

Take bundle charts ${\Phi }_i:E{|}_{U_i}\to U_i\times {\mathbb{R}}^r$ with transition functions $g_{ij}:U_i\cap U_j\to GL(r,\mathbb{R})$. Over $f^{-1}(U_i)$ defineTheir transition functions are $g_{ij}\circ f$, so they give the fibre producta smooth manifold and make its projection to $M$ a rank-$r$ pullback vector bundle.

A local section $s:U\to E$ determines the sectionof $f^{\ast }E$ over $f^{-1}(U)$. It is unique with second component $s\circ f$.

A covariant derivative is an $\mathbb{R}$-linear mapsatisfying $d_A(fs)=df\otimes s+f{d}_{A}s$. It is local: its value over an open set depends only on the restriction of the section there. A horizontal connection differentiates a section, projects its derivative vertically, and identifies the vertical tangent with the fibre.

In a local frame write $d_{A}s=ds+A_{i}sd{y}^i$. If $x^a$ are coordinates on $M$, the defining identity in the question forceswithThus $A^{\prime }=f^{\ast }A$, the pullback connection.