I have a beginner question in Riemannian geometry. Given a manifold $\mathcal{M}$ endowed with a Riemannian metric $g$, I assume this metric is represented locally by a family of positive definite matrices $(A_q)_q$ in local coordinates.
I assume I have an inequality of the type : $A_q \geq M^{-1}_q$ near every $q \in \mathcal{M}$, where $M_q$ is a positive definite matrix and the inequality is in the sense of Loewner order.
Is there anything intelligent to say about the $g$-geodesic distance ? For example, a link between the $g$-geodesic distance and the $M$-geodesic distance or an integral linked with $M$ ?
My general question is whether it is possible to turn a local comparison of a metric/cometric (that appears in the Hamiltonian equations of geodesics) into a global result on the geodesic distance ?
EDIT : I forgot the inverse on $M_q$.