Timeline for Bochner Laplacian in coordinates

8 events

when toggle format	what		by	license	comment
Mar 4 at 18:52	vote	accept	B.Hueber
Mar 4 at 18:51	comment	added	Willie Wong		(The previous comment makes more sense if you denote $\nabla$ in a graded way, so $\nabla^{(k)}$ acts on $T^*M^{\otimes k}$. )
Mar 4 at 18:50	comment	added	Willie Wong		Yes, for the (co)tangent bundle you the action of $\nabla$ on $T^M^{\otimes k}$ is the tensor product connection of $\nabla$ acting on $T^M$ with the connection $\nabla$ acting on $T^M^{\otimes (k-1)}$. It just happens that knowing $\nabla$ on $T^M$ you can build the rest inductively and abuse notation to call them all $\nabla$.
Mar 4 at 18:46	comment	added	B.Hueber		I see, so this notational ambiguity is somehow specific to the tangent bundle. Thanks!
Mar 4 at 18:33	comment	added	Willie Wong		@B.Hueber ^^ yes. It is helpful to note that when $\nabla$ is a linear connection on the bundle $E$, and $f$ a section, then $\nabla f$ is a section of $T^M \otimes E$ and it doesn't make sense to take "$\nabla \nabla f$ since $\nabla$ is not defined to act on the bundle $T^M \otimes E$.
Mar 4 at 18:20	comment	added	B.Hueber		I am aware of the fact that the lecture notes are written for generic bundles. But the same local formula holds true when taking the tensor bundles $T^{\ast}M^{\otimes_{k}}$ and the Levi-Civita connection as a special case
Mar 4 at 18:19	comment	added	B.Hueber		Could you have a look at my edit? Is this also correct?
Mar 4 at 18:01	history	answered	Willie Wong	CC BY-SA 4.0