If I understand things correctly, $\Delta_g$ is a non-negative Laplace operator. Are you assuming that $q\ge 0$?

@Iosif Pinelis The last question (about $r(t,s)$) is asking about $\exp r$ being positive-definite when $r$ is positive definite. Since $r$ can be written as the sum of a symmetric matrix and a skew-symmetric matrix, and moreover since the positiveness assumption does not concern the skew-symmetric part, you arrive at the question formulated in my answer ($A,S$ business).

I did not intend to edit your text, but for some strange reason, I landed in your tex file. I erased my intervention, sorry about that.