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Topic: Riemannian Metric Topology
Replies: 10   Last Post: Apr 4, 2012 3:52 PM

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 Jeff Rubin Posts: 17 Registered: 11/25/07
Re: Riemannian Metric Topology
Posted: Apr 1, 2012 10:49 PM

On Sunday, April 1, 2012 3:45:47 PM UTC-7, 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 p189-190
> > 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 pseudo-metric 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
> > non-circularly 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.

Date Subject Author
3/31/12 Jeff Rubin
3/31/12 HOPEINCHRIST
4/1/12 Stuart M Newberger
4/1/12 Jeff Rubin
4/2/12 Stuart M Newberger
4/3/12 Jeff Rubin
4/3/12 M. Vaeth
4/4/12 M. Vaeth
4/3/12 Stuart M Newberger
4/4/12 Guest
4/4/12 Stuart M Newberger