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 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.

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.