Abstract: The unit circle viewed as a Riemannian manifold has diameter (not 2 but rather) π, illustrating the difference between intrinsic and ambient distance. Gromov proceeded to erase the difference by pointing out that when a Riemannian manifold is embedded in L∞, the intrinsic and the ambient distances coincide in a way that is as counterintuitive as it is fruitful. Witness the results of his 1983 Filling paper. Gromov exploited this embedding to prove a universal upper bound for the systole of an essential (e.g., aspherical) manifold, and created an entirely new area of research around the invariants called filling radius and filling volume, which is active until today with recent contributions by Larry Guth and others.
In the context of the manifold of nonsingular matrices, Asaf Shachar asked when the intrinsic and the ambient (Euclidean) metrics are bilipschitz equivalent. The answer turns out to hinge on the structure of the stratification of the determinantal variety.