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Let $\pi:P\rightarrow M$ is a principal $G$ bundle. I want to introduce the notion of connection as a way to uniquely lift the structures on $M$ to structures on $P$, namely vector fields and paths.

The notion of lifting of a structure is common in mathematics.

  1. Given a continuous map $f:X\rightarrow Y$, and a path $\gamma:[0,1]\rightarrow Y$, the question of whether it can be lifter to a path on $X$. Same is discussed for homotopy lifting and lifting of arbitrary maps.
  2. An $R$-module $P$ such that for each surjective morphism of $R$-modules $N’\rightarrow N$, any morphism of $R$ modules $P\rightarrow N$ can be lifted to an $R$module morphism $P\rightarrow N’$ is special and it also has a name; projective module.

In similar way, one can ask the question : given a structure on $M$, say a vector field $X:M\rightarrow TM$, can one lift it to a vector field on $P$? We can give a set level map $\tilde{X}:P\rightarrow TP$ as $p\mapsto \tilde{X}(p)$ where $\tilde{X}(p)$ is some element in the inverse image of $X(m)$ under the map $\pi_{*,p}:T_pP\rightarrow T_mM$. It is not obvious why this $\tilde{X}$ is smooth and it is clearly not uniquely defined. Connection on principal bundle fix this issue, by giving unique vector field (horizontal) on $P$ given a vector field on $M$. Same is true for paths on $M$. Because of local trivialisation, one can lift the path. But uniqueness is not clear. Connection fix this issue, given a path $\alpha:[0,1]\rightarrow M$, fixing a point in the fibre of $\alpha(0)\in P$, there exists unique lift of $\alpha$ in $P$ starting at the point specified.

Do we lose something if this is given as motivation to introduce the notion of connection? I want to motivate connection in this way to a group of graduate colleagues. Is this reasonable?

First thing to observe after defining Connection on principal bundle as splitting of Atiyah sequence $$0\rightarrow (P\times \mathfrak{g})/G\rightarrow (TP)/G\rightarrow TM\rightarrow 0$$ is that connection is giving a way to lift a vector field on $M$ to a vector field on $P$. So, I think this is reasonable motivation to introduce connection. Any comments are welcome.

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  • $\begingroup$ I am not very sure if I have written down clearly, may be I am assuming something but didn’t mention here. I am not 100% sure if this is suitable here, but I will leave it as it is.. if users think I should move to MSE, I am ok with it,, $\endgroup$
    – Praphulla Koushik
    Commented Oct 18, 2019 at 18:29
  • $\begingroup$ One can certainly introduce the notion of connections (not just on principle bundles, but more generally on fibred manifolds) through lifts. I know that this is discussed in Kobayashi-Nomizu. I have some notes which may or may not be relevant for your discussion (since I don't really care about $G$ bundles there). $\endgroup$
    – Willie Wong
    Commented Oct 19, 2019 at 15:40
  • $\begingroup$ @WillieWong it is discussed in Kobayashi and Nomizu but it was not introduced in that way... so, looking for some suggestions on what I lose if I introduce in this way.. $\endgroup$
    – Praphulla Koushik
    Commented Oct 20, 2019 at 2:36
  • $\begingroup$ You don't lose anything. Look at the notes I linked to. On the level of fibred manifolds, a connection is pretty much equivalent to a horizontal distribution (see Definitions 12 and 13). On $G$ bundles you would want the distribution to be $G$ invariant in the obvious way. $\endgroup$
    – Willie Wong
    Commented Oct 20, 2019 at 2:43
  • $\begingroup$ @WillieWong I am reading that notes. It looks good to me :) thanks for sharing... I will ask if I have any specific questions $\endgroup$
    – Praphulla Koushik
    Commented Oct 20, 2019 at 3:05

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