
Riemannian Metric Topology
Posted:
Mar 31, 2012 11:12 AM


This is a question about a step in a proof that appears in two of my textbooks:
Lang[1999] Fundamentals of Differential Geometry, Serge Lang AMR[1988] Manifolds, Tensor Analysis, and Applicxations, second edition, R. Abraham, J.E. Marsden, T. Ratiu
The setting is that we have a connected Hausdorff manifold, X, and a Riemannian metric, g, on X. No other assumptions are made about the manifold, so in particular I don't know that it is paracompact, normal, regular, or if it admits partitions of unity. I don't even know if the Hilbert space (or spaces) it is modeled on is separable or not. For simplicity, I assume X us a manifold without boundary.
Given a point x of X and a chart (U, \phi) at x for X, where \phi(U) is open in a Hilbert space E, one easily gets the positive definite, invertible, symmetric operator A(x) on E which corresponds to g(x). Given an element z of the tangent space above x, one also easily gets the real value (g(x)(z, z))^{1/2}. We then go on to define a length function L_g which assigns a real number L_g(\gamma) to each piecewise C^1 path \gamma:J=[a,b] \to X as follows:
L_g(\gamma) = \int_J (g(\gamma(t))(\gamma'(t),\gamma'(t)))^{1/2} dt.
We then define a function dist_g: X x X \to R by
dist_g(x,y)=inf{L_g(\gamma) : \gamma is a piecewise C^1 path in X from x to y, defined on the closed interval J=[a,b]}
Without any difficulty, dist_g is a pseudo metric. However, we have not yet shown that the topology it induces on X is the same as the original manifold topology. The first main point of the proofs in both books (Lang p189190 and AMR p381 Proposition 5.5.10) is to show that dist_g is not just a pseudo metric but is in fact a metric. So we start with distinct points x and y of X and set out to show dist_g(x,y) > 0. We have the chart (U, \phi) at x, as above, and we can arrange U to be small enough that y is not in U, since the manifold is assumed to be Hausdorff. Working in \phi(U) we find an r>0 such that the closed ball D(\phi(x),r) is contained in \phi(U) and such that certain other properties hold. Let S(\phi(x),r) be the boundary of D(\phi(x),r). Then we define D(x,r)=\phi^{1}(D(\phi(x),r)) and S(x,r)=\phi^{1}(S(x,r)), both subsets of U.
Since \phi is a homeomorphism, D(x,r) and S(x,r) are closed in U (not necessarily closed in X). To me, this is a key stumbling point, as I'll explain. We next let \gamma:J \to X be any piecewise C^1 path in X from x to y. Both proofs make the following assumption: since x is in D(x,r) and since y is not in U, the path \gamma must cross S(x,r). Neither author explicitly proves this assumption (and AMR doesn't even state it).
When I set out to prove this, using the continuity of \gamma and the connectedness of J, I quickly run into the need to show that D(x,r) is closed in X, not just in U. If X were known to be regular, it would not be a problem to take r small enough that D(x,r) was closed in X. But as I mentioned at the beginning, I don't know that X is regular. If I could show that the pseudometric topology for X induced by dist_g was the same as the original manifold topology, I would also get that X was regular. But I don't see how to do that without first completing the first part of the proof.
The whole question seems to be, can I make r small enough that D(x,r) stays away from the topological (in the original manifold topology of X) boundary of U? But this does not seem to be a local issue, since it depends on what is closed in X which in turn, depends on what is open everywhere in X including outside of U.
In Abraham's "Foundation of Mechanics", I found a statement to the effect that a manifold which admits a Riemannian metric is necessarily second countable. However, I don't see how that could be applied here (nor do I immediately see why it is true).
Now, assuming that the statements that the authors are trying to prove is actually true, then X will turn out to be a metric space and therefore regular. So how do I get this regularity early enough in the proof to noncircularly use it to show \gamma must cross S(x,r)? Alternatively, how do I directly show that \gamma must cross S(x,r)?
Interestingly, a third reference I have (Kobayashi & Nomizu, Foundations of Differential Geometry, Volume 1, 1963 and 1991) gives a proof (Chapter IV Proposition 3.5, p166) which doesn't seem to proceed the same exact way, but it is completely impenetrable.

