Let $(G,\circ)$ be a Lie group, where $G\subseteq\mathbb R^n$ is a compact, smooth manifold. I call the neutral element $e$.
I figured that there exists a bi-invariant metric on $G$ and I also figured that then one parameter subgroups are geodesics.
Is for every $p\in G$, $v\in T_eG$ and sufficiently small $a,b\geq 0$ the curve $$ \gamma: [-a,b] \to G,\qquad t\mapsto p\circ\exp(t v)$$ the shortest curve on $G$ that connects $\gamma(-a)$ and $\gamma(b)$ with respect to the Euclidean norm on $\mathbb R^n$?