Timeline for Definition of Connection as G-invariant splitting of a sequence which is a pulled back sequence of bundles

4 events

when toggle format	what		by	license	comment
Sep 19, 2016 at 19:49	vote	accept	Raul
Sep 18, 2016 at 11:34	comment	added	Ben McKay		No. We ARE making things descend. They even emphasize in italics that this is an exact sequence on $M$. You start with the sequence on $P$: $0 \to T_F P \to TP \to \pi^{-1}TM$. You then look at the $G$-invariant sections of those bundles over each point of $M$, to construct bundles on $M$: $0 \to ad(P) \to E(P) \to TM \to 0$. So $ad(P)$ means the bundle on $M$ whose local sections over an open subset $U_M \subset M$ are the $G$-equivariant local sections of $T_F P$ defined on the set $U_P=\pi^{-1}U_M$.
Sep 18, 2016 at 8:10	comment	added	Raul		Thanks Ben. I'm not sure I understood the last bit, I understand that there is a 1-1 correspondence between sections of the vector bundle and equivariant functions on $P$. But here we are not making things descend to $M$ but to pullback bundles from $M$ to $P$. After a long though I came to the conclusion that we need the $G$-invariant vector fields because the requirement of the $G$-invariant splitting implies that the vector fields of $TP$ must be $G$-invariant (this follows directly from what I wrote above $gv_p = X_{pg} + H_{pg} = v_{pg}$). Please let me know if you agree with this
Sep 17, 2016 at 10:55	history	answered	Ben McKay	CC BY-SA 3.0