effect of global topology on analytic continuation over a manifold

[drDave]

Suppose \(\displaystyle M\) is a connected analytic manifold with metric \(\displaystyle g\) which is everywhere analytic. Define \(\displaystyle \gamma(g(x))\) as the germ of the metric at the point \(\displaystyle x \in M\).

Starting from a single germ \(\displaystyle \gamma\), I wish to use analytic continuation to determine the metric \(\displaystyle g\) at every point \(\displaystyle x \in M\).

QUESTION: What additional topological assumptions about \(\displaystyle M\), assumptions about \(\displaystyle g\), and / or restrictions on \(\displaystyle \gamma\) are necessary and / or sufficient to say that this is possible?

 

[drDave]

Here are some of my ideas, which I hope make sense, but please tell me if they do not!

Observation 1: If the genus of \(\displaystyle M\) is zero, a solution to \(\displaystyle g\) should be possible everywhere in \(\displaystyle M\) for any arbitrary initial germ \(\displaystyle \gamma\).
Observation 2: If every component of \(\displaystyle \gamma\) is zero (the "flat" germ), then \(\displaystyle M\) must be everywhere flat, so the genus of \(\displaystyle M\) must be zero.
Observation 3: If the genus of \(\displaystyle M\) is greater than zero, then certain restrictions must be placed on \(\displaystyle \gamma\), e.g. it cannot be the "flat" germ. Probably more restrictions too, although I'm not sure how to characterize them.

Continuing the train of thought from the above, given two germs \(\displaystyle \gamma_1\) and \(\displaystyle \gamma_2\) we can ask: does there exist any manifold and metric pair \(\displaystyle (M,g)\) which is inhabited by both germs simultaneously? (By inhabit simultaneously I mean: \(\displaystyle \gamma_1\) is the germ of \(\displaystyle g\) at \(\displaystyle x_1\), \(\displaystyle \gamma_2\) is the germ of \(\displaystyle g\) at \(\displaystyle x_2\), with \(\displaystyle x_1, x_2 \in M\).) If yes, we call them equivalent: \(\displaystyle \gamma_1 \approx \gamma_2\). In this way, an equivalence relation could be established over the space of all germs. As an example, the "flat" germ would be equivalent only with itself.

Does this make sense? I would be especially interested to know whether the equivalence relationship I describe makes sense or has even been explored. Any thoughts would be appreciated.