Integrating Metric Tensors: Conditions for Obtaining a Global Metric Function

[Ben Niehoff]

Under what conditions can the metric tensor be integrated to a global metric function? i.e., a function g(x,y) that gives the distance along a geodesic between x and y?

For example, we can do this on the sphere using spherical trigonometry (cf. http://en.wikipedia.org/wiki/Spherical_law_of_cosines). It looks like we can also do this on a surface of constant negative curvature.

Some obvious conditions are that a geodesic must exist between all pairs (x,y), and it must be unique (possibly with some restriction such as the spherical example above). These conditions are necessary, but are they sufficient?

 

[wofsy]

You get a metric by saying that the distance between two points is the infimum of the lengths of all piece wise smooth curves connecting the points.

If the space is complete under this metric then there will always be a length minimizing curve that is a geodesic.

If it is not complete this may not be true. For instance, the plane minus the origin with the Euclidean metric. The points (1,0) and (-1,0) are not connected by a straight line and in fact there is no curve that minimizes the length of all curves between them. But the infimum of all of the lengths is 2.