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I am trying to solve the following exercise:

Let $P$ be a vector bundle over a smooth manifold $M$ with a connection $\nabla$, and let $p \in M$ . Show that there is an open set $U$ of $M$ with $p \in U$ and a frame $E_1, \ldots, E_k$ of $P$ defined in $U$ such that for all $v \in T_pM, \nabla_v \ E_i = 0$, for $i \in \{1, \ldots, k\}$.

However, I am not sure how to begin, and I would like some sort of hint, if possible - how would one define such a frame? Is there some intuitive way of doing it?

Thank you in advance.

essay
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1 Answers1

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Work over a coordinate patch $U$ around $p$ with local coordinates $\phi:U\xrightarrow{\sim\,}\Bbb R^n$ ($n$ is the dimension of $M$). Take a basis $(e_1,\dots,e_r)$ a basis of the ($r$-dimensional) vector space $P_p$. Extend it to a local basis of $P|_U$ by the formula $$\forall q\in U,\,E_i(q)=\mathrm{PT}^\nabla_{\gamma_{q}}(e_i)$$ Where $\mathrm{PT}^\nabla_\gamma$ is parallel transport (in the sense of $\nabla$) along a path $\gamma$, and $\gamma_{q}:[0,1]\to M$ is the path connecting $p$ to $q$ defined by $$\phi(\gamma_{q}(t))=\phi(p)+t(\phi(q)-\phi(p))$$ $(E_1,\dots,E_r)$ form a basis of sections of $P$ over $U$, and you can check (almost by definition) that $\left(\nabla E_j\right)_p=0$.

Olivier Bégassat
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  • Can you point out a reference in which I can find a proof of existence of parallel transport in an arbitrary vector bundle? (So far I've seen it only for the tangent bundle.) And thank you for your answer. – essay Jan 04 '15 at 22:49
  • @sylvia You'll find it in these notes: http://webusers.imj-prg.fr/~vincent.minerbe/m2dg.pdf (page 41 and onwards). The gist is that parallel transport along a fixed curve obeys a first order linear differential equation, and thus exists and is smooth in the parameters (i.e. the curve and the starting vector). – Olivier Bégassat Jan 04 '15 at 22:54