Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and identity element $e$. I want to define a Riemannian metric on $G$ such that the curves $\gamma_{X, g}(t) = \exp(tX)\cdot g$ with $t\in \mathbb{R}$ are geodesics for any $X\in\mathfrak{g}$ and $g\in G$, where $\exp$ here is the usual Lie-theoretic exponential map---i.e., the flow along the integral line through $e$ of the left-invariant vector field whose value at $e$ is $X$. My questions are the following:
- What properties does $G$ need to have in order for such metric to exist?
- Assuming those conditions are met, how much does that constrain the metric?
This is a small modification to another question, whose assumptions turned out to be too weak to lead to any interesting constraints on the metric in general; see Lee Mosher's answer there. Does this slightly stronger assumption allow me to make more concrete statements about the metrics that would satisfy this?