Can we lift paths of a Lie group quotient $G\to G/H$?

Question

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{\gamma}$on $G$ such that $\pi\circ\tilde{\gamma}=\gamma$?

For each $t\in[0,1]$, we can write $\gamma(t)=\tilde{\gamma}(t)H$, where $\tilde{\gamma}(t)\in G$. But the question is if we can choose $\tilde{\gamma}$ smoothly?

Answer 1

You can lift such paths. The projection $p:G\rightarrow G/H$ defines a principal $H$-bundle over $G/H$, take any Ehresmann connection on this bundle i.e a distribution on $G$ of dimension $dim G/H$ supplementary to the distribution tangent to the fibres and invariant by $H$, then you can lift any path $\gamma:[0,1]\rightarrow G/H$ to an horizontal path $\tilde\gamma:[0,1]\rightarrow G$.

https://en.wikipedia.org/wiki/Ehresmann_connection#Parallel_transport_via_horizontal_lifts