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 one-parameter subgroups $\gamma_X(t) = \exp(tX)$ with $t\in\mathbb{R}$ are geodesics for any $X\in\mathfrak{g}$, 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, is the metric unique?
I know that if the metric is bi-invariant (i.e., both left- and right-invariant) under the group action, then $\gamma_X(t) = \exp(tX)$ are geodesics, and if $G$ is compact, such a metric can be defined. But does it also go the other way around? Namely, do my requirements imply that the metric has to be bi-invariant, and the group has to be compact? I'd appreciate it if anyone could shed any light on this!
Edit: As per Lee Mosher's answer below, the condition I listed in the first paragraph is still too weak for me to make any concrete general statements about the metrics that satisfy it. A slightly stronger requirement would be to demand that not only $\gamma_X(t) = \exp(tX)$, but also $\gamma'_X(t) = \exp(tX)g$ be a geodesic for every $g$ in the group and $X$ in the Lie algebra. Is there anything interesting that can be said about that case?
Edit 2: The question made in the previous edit is now in a new post, as Lee Mosher's answer below satisfactorily addresses the conditions of the question as it was originally posed.
Best Answer
For a simple counterexample take the Lie group $\mathbb R^2$ with two metrics. The first is the usual Euclidean metric, which can also be written in polar coordinate form $$ds^2 = dx^2 + dy^2 = dr^2 + r^2 d\theta^2 $$ The second, when written in polar coordinates, has the form $$ds^2 = dr^2 + \sinh^2(r) d\theta^2 $$ which is isometric to the hyperbolic plane $\mathbb H^2$ (see this math.stackexchange question). But compactness fails, and the second metric is not bi-invariant.
Perhaps you can see from this example why your questions don't really have much hope: requiring just that the one-parameter subgroups are geodesics is a very loose requirement, allowing a lot of play in the construction of the Riemannian metric. In this example, all I had to do was tweak the factor in front of $d\theta^2$.