Home > Intrinsic metric

Metrics which satisfy the above property are called intrinsic. What follows next is the formal mathematical way to describe it.

1 Definition and discussion

A metric space is called length space or path metric space or equivalently the metric is called intrinsic if the distance between any pair of points in is equal to the infimum of lengths of curves connecting these points. Equivalently is intrinsic if for any and any pair of points there is such that and are smaller then .

The Hopf-Rinow theorem states that if a length space is complete and locally compact then any two points in can be connected by minimizing geodesic and any bounded closed sets in are compact. It is due to Heinz Hopf and his student Willi Rinow.

Given any metric , one can define the induced intrinsic metric by saying is to be the infimum of lengths of paths connecting and (or if there is no contractible path connecting and ). Clearly

.

In general the topology defined by can be coarser than the one defined by .

2 Examples

• Euclidean space.
• Riemannian manifold.
• Finsler manifold.
• sub-Riemannian manifold.