
Re: Riemannian Metric Topology
Posted:
Apr 1, 2012 10:49 PM


On Sunday, April 1, 2012 3:45:47 PM UTC7, smn wrote: > On Mar 31, 8:12 am, Jeff Rubin <JeffBRu...@gmail.com> wrote: > > 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. > > Hello , Unless you assume the manifold topology is Hausdorf you will > not be able to show that the pseudo metric is a metric with the same > topology since a metric space is Hausdorf.It still may be the case > that the pseudo metric topology is the same as as the manifold > topology anyway.Maybe an almost same proof works for this ,I havent > tried . Good Luck smn .
Yes, I did stipulate early in the post that the original manifold topology was assumed to be Hausdorff.

