Which vector bundle are the Christoffel symbols sections of? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:07:27Z http://mathoverflow.net/feeds/question/87524 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87524/which-vector-bundle-are-the-christoffel-symbols-sections-of Which vector bundle are the Christoffel symbols sections of? Qfwfq 2012-02-04T13:10:32Z 2012-02-04T13:42:24Z <p>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 $M$.</p> <blockquote> <p>Is there a vector bundle, naturally associated to $M$, of which the collection of Christoffel symbols represents a section?</p> </blockquote> http://mathoverflow.net/questions/87524/which-vector-bundle-are-the-christoffel-symbols-sections-of/87528#87528 Answer by Robert Bryant for Which vector bundle are the Christoffel symbols sections of? Robert Bryant 2012-02-04T13:33:38Z 2012-02-04T13:33:38Z <p>No, the Christoffel symbols are not the components of a section of a natural <em>vector</em> bundle over $M$. Rather, they are the components of a section of a natural <em>affine</em> 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.</p> http://mathoverflow.net/questions/87524/which-vector-bundle-are-the-christoffel-symbols-sections-of/87530#87530 Answer by BS for Which vector bundle are the Christoffel symbols sections of? BS 2012-02-04T13:42:24Z 2012-02-04T13:42:24Z <p>Connections on a vector bundle $E\to M$ are sections of an <em>affine</em> bundle associated to $E$.</p> <p>Namely there is a vector bundle $J^1E$ of $1$-<a href="http://en.wikipedia.org/wiki/Jet_%28mathematics%29" rel="nofollow">jets</a> 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").</p> <p>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.</p>