Let $P(M,G,\pi)$ be a principal bundle. Let $\mathcal{H}\subseteq TP$ (equivalently $\omega:P\rightarrow \Lambda^1_{\mathfrak{g}}T^*P$) be a connection on the principal bundle.
Proposition $1.2$ (page $65$) of Kobayashi and Nomizu says the following:
Given a connection on $P$ and a vector field $X$ on $M$, here is a unique horizontal lift $X^*$ (a vector field on $P$) of $X$.
Proposition $3.1$ (page $69$) of Kobayashi and Nomizu says the following:
Let $\tau=x_t, 0\leq t \leq 1$ be a curve of class $C^1$ in $M$. For an arbitrary point $u_0$ of $P$ with $\pi(u_0)=x_0$, there exists a unique lift $\tau^*=u_t$, of $\tau$ that starts from $u_0$.
So, a connection on a principal bundle lifts vector fields on $M$ uniquely to (horizantal) vector fields on $P$ and lifts paths on $M$ uniquely (after fixing initial point) to horizantal paths on $P$.
Does connection on $P(M,G)$ lift any other geometric structures on $M$ to structures on $P$?
I am not even sure if this question is well posed. Nevertheless, I will leave it like this, I can take suggestions to make this better.