Ben Niehoff

Science Advisor

Gold Member

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## Main Question or Discussion Point

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?

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?