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The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they don't correspond to a section of a tensor bundle over $M$.

Is there a vector bundle, naturally associated to $M$, of which the collection of Christoffel symbols represents a section?

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Yes, it's naturally a section of a vector bundle over not $M$ but over the frame bundle (the bundle of, say, orthonormal bases of $T_xM$) of $M$. It's not possible that it be a section of a vector bundle over $M$, because the Christoffel symbols depend on not just a point on $M$ but also a choice of co-ordinates (or at least an infinitesimal choice which is what a frame of tangent vectors effectively is). –  Deane Yang Feb 4 '12 at 13:32
If $\nabla$ is a connection on a vector bundle $V \to M$ then one often writes in local coordinates $\nabla_{\partial_i} = \partial_i + \Gamma_i$ where $\Gamma_i$ is a section of the bundle $End(V)$. Regarded as a $n \times n$ matrix, the $(j,k$ entry of $\Gamma_i$ is the Christoffel symbol $\Gamma_{ij}^k$. I don't think you can do better than that because $\nabla_X$ does not depend $C^\infty(M)$-linearly on $X$. –  Paul Siegel Feb 4 '12 at 13:45

2 Answers 2

No, the Christoffel symbols are not the components of a section of a natural vector bundle over $M$. Rather, they are the components of a section of a natural affine bundle over $M$, namely the connection bundle $C(M)$, which has the bundle $TM\otimes T^\ast M\otimes T^\ast M$ as its associated vector bundle.

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Connections on a vector bundle $E\to M$ are sections of an affine bundle associated to $E$.

Namely there is a vector bundle $J^1E$ of $1$-jets of sections of $E$, and an exact sequence of bundles $$0\to T^*M\otimes E\to J^1E \overset{p}\to E\to 0$$ where the map $p$ is the evaluation ("$0$-jet").

Then a connection is a section of the affine bundle of sections (sic) of $p$, namely the $s\in Hom(E,J^1E)$ such that $p\circ s=id_E$. The associated vector bundle is $Hom(E,T^*M\otimes E)\simeq T^*M\otimes End(E)$, where one can view the Christoffel symbols (if $E=TM$) as living : once (local) a trivialisation is chosen there is an associated "trivial" connexion, and any other connection differs from it by a section of this vector bundle.

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