Let $G$ be a Lie group and $\alpha: [0,1] \to G$ a smooth path, connecting the neutral element $n_G$ of the group with a group element $g$, i.e. $\alpha(0)=n_G$ and $\alpha(1)=g$.
Can we find a homotopic one-parameter subgroup $\gamma: {\Bbb R} \to G$ connecting the neutral element of the group with the group element $g$, i.e. $\gamma(0) = n_G$ and $\gamma(1) = g$, and $\gamma$ restricted to $[0,1]$ is in the same homotopy class as $\alpha$?
If I have a homotopy class for connecting a group element to the neutral element, can I chose a representative path in this homotopy class to be a one parameter group?
Assume $G$ is compact with a bi-invariant Riemannian metric; then one-parameter subgroups are exactly geodesics containing the origin [e.g. do Carmo, Riemmanian Geometry, Chapter 3, Exercise 3b]. Then the existence of a geodesic within a path homotopy class follows by finding a path of minimal Lipschitz constant within the homotopy class of $\alpha$.