Let $(M,g)$ be a Riemannian manifold. Let $\nabla$ be a connection for $M$. Let $c(.)$ be a $\nabla$-geodesic. I believe I can also construct a new distance function $d_{g,\nabla}(x,y):=\int_0^1 \|\dot{c}(t)\|_g dt$, where $c(.)$ is the shortest $\nabla$-geodesic connecting $x$ to $y$ and $\|.\|_g$ is the Riemannian norm induced from the Riemannian metric $g$.
I can also take any $\nabla$-geodesic $c(.)$ and parameterize it so that $\|\dot{c}(t)\|_g$ is constant.
Here is my question. What kind of object is $(M,g,\nabla)$? Is there a name for this new distance function $d_{g,\nabla}(.,.)$? I was hoping there are some resources out there for Riemannian manifolds equipped with non-metric connections.