Glossary of Riemannian and metric geometry

The geodesic metric space is a term from mathematics. He describes spaces in which one can find for every two points a shortest curve. The notion generalizes the concept of complete Riemannian manifolds to general metric spaces.

Geodesics in metric spaces

Be a metric space. A curve is a continuous mapping, which is a closed interval in. The length of the curve is defined as

From the triangle inequality, the inequality follows. The curve is called minimizing geodesic if equality

Applies.

Definition

A metric space is called geodesic if for any two points a minimizing geodesic with

There.

Set of Hopf Rinow

For a Riemannian manifold to define a metric by

For. It goes through all piecewise differentiable curves and connect, and denotes the Riemannian length of which in accordance with

Is defined. Thus the Riemannian manifold into a metric space.

From the set of Hopf Rinow follows:

  • Is a geodesic metric space

If and only if one of the following equivalent conditions is satisfied:

  • The Riemannian manifold is geodesically complete,
  • There exists a so that the exponential map is defined for all
  • The metric space is complete as a metric space.

Counterexample

Be

The dotted complex plane with the metric

For.

Then, for example, or, in two cases the pairs of points can not be connected by curves of length 2.

Interval (mathematics)
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