Premetric that induces the Lie group conditional flow field

Determine whether there exists a premetric on a Lie group G (equipped with a left-invariant Riemannian metric) whose gradient with respect to the first argument induces the conditional vector field u_t(g | g_1) = (L_g)_* log(g^{-1} g_1) / (1 - t) used in Lie group Flow Matching, and ascertain conditions under which the logarithmic distance (the length of the exponential curve between g and g_1) induces this field beyond the bi-invariant metric case where geodesics and exponential curves coincide.

Background

Riemannian Flow Matching constructs conditional vector fields by differentiating a premetric, often chosen as the geodesic distance or a spectral approximation. In this work, the authors propose an intrinsic Lie group formulation where conditional integral curves are exponential curves, leading to the specific vector field u_t(g | g_1) = (L_g)_* log(g^{-1} g_1)/(1 - t).

They investigate whether their Lie group construction can be recovered as a special case of Riemannian Flow Matching by appropriately choosing a premetric. The obvious candidate, the logarithmic distance (length of the exponential curve), only reproduces the desired conditional field in specific situations, such as when the Riemannian metric is bi-invariant, making geodesics and exponential curves coincide. This leaves open the existence or characterization of a general premetric that induces the proposed conditional flow field.