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In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb R^k$ then the holonomy group $\mathrm{Hol}(\nabla)$ comes equipped with a natural representation on $\mathbb R^k$, or equivalently, with an embedding of $\mathrm{Hol}(\nabla)$ in $\mathrm{GL}(k,\mathbb R)$. He calls this representation the holonomy representation.

But actually $\mathrm{Hol}(\nabla)$ is defined as a subgroup of $\mathrm{GL}(k,\mathbb R)$ up to a conjugation. Does it mean that actually the holonomy group comes equipped with a family of representations on $\mathbb R^k$ and "the holonomy representation" is actually a class of representations?

On the other hand, it's clear that the holonomy group $\mathrm{Hol}_x(\nabla)$ with fixed basepoint comes equipped with a natural representation on $E_x$, the fibre of $E$ over $x$.

Appliqué
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    I don't know the book you mention in your question, but to me, when you have a group ($Hol(\nabla)$) and two matrix representations of dimension k for that group, and you know that those two matrix representations are conjugate, then they are essentially the same, i.e. they have the same representation-theoretic properties. So you can very well call these conjugate representations "the holonomy representation". – jflipp Oct 28 '14 at 22:53

1 Answers1

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Yes. The holonomy representation is an isomorphism class of representations.

Qiaochu Yuan
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