I just came upon the definition of a metric tensor in Riemannian geometry as a way to define local distances. As I understand it, metric tensors are defined at every point on a Riemannian manifold, but vary smoothly on the manifold structure. Different paths between two points might have different lengths, and the distance between two points is defined as the infimum of all possible lengths.
My question is the following : can the length of a loop around two points (A, B) on a Riemannian manifold be inferior to the distance between these two points ? I've sketched the situation below in dimension 2.
Trying to formalize it, can : 
?
where A and B are points, P is a path which is a loop going around A and B, and g would be the local notion of distance. I'm not sure whether the integral is the proper way to write it.
Thanks
Yes, on a Riemannian manifold, it's definitely possible for the length of the loop to be smaller than the distance from $A$ to $B$.
Here's an example. Let $M$ be $S^2$ with the south pole removed (so $M$ is diffeomorphic to $\mathbb R^2$), endowed with the restriction of the standard round metric. Let $A$ and $B$ be two antipodal points on the equator, and let $P$ be a latitude circle very close to the (missing) south pole. Then $A$ and $B$ are "inside" $P$ by any reasonable interpretation of what "inside" means, but the length of $P$ can be made as small as you like by taking it close enough to the south pole.