Re: Topological Metric Space vs. Metric on a Manifold

On Sat, 8 Dec 2007 19:32:05 -0500, "David Park" <djmpark@xxxxxxxxxxx>
fed this fish to the penguins:

If I use a distance function to make R^n into a 'Metric Space' and then use
this to make charts for a manifold,

What do you mean by this?

does that imply that there is any kind
of metric on the manifold? I say the answer is no, it only gives us a
concept of nearness and continuity on the manifold and it would take extra
structure to put a metric on the manifold.


Correct, a manifold structure does not imply the existence of a
metric. Or more correctly, one can prove that it is possible to come
up with a metric on a manifold that is compatible with the topology
(with the usual definition of manifold that entails countability
assumptions) but this metric is highly non-unique, non-canonical, and
always involves some arbitrary choices.

By the way, this is a purely topological result, the differentiable
structure is irrelevant for the matter.

So when someone says that such a space is a metric space, it (almost)
always implies a *choice* of an *extra* structure in terms of a
definite metric that has a special relevance for this or that problem.

Best regards,
G. Rodrigues

Note: one must be careful with the terminology here. A Riemannian
structure is also called a metric in many quarters. But the same
answer stands: while it is possible in every manifold to put a
Riemannian structure, it is non-unique, non-canonical and always
involves some arbitrary choices.
.

Relevant Pages

  • Re: Contractible metric space
    ... In any metric space, ... Recall that a Riemannian manifold is a manifold with a Riemannian ... and induces the quotient topology. ...
    (sci.math)
  • Re: Question in Analysis
    ... >> metric space X and D a subset of a metric space Y. It is a basic ... >> theorem that the mapping f is continuous iff for every open set A in Y ... > to the assertion "for every open set A in D the inverse image of A is ... manifold M, for a particular chart, the set xis not ...
    (sci.math)
  • Re: integrals with values in a metric space???
    ... M is a metric space, ... on curves in the manifold, ... allows line integrals (e.g. of real- ...
    (sci.math)
  • Topological Metric Space vs. Metric on a Manifold
    ... If I use a distance function to make R^n into a 'Metric Space' and then use this to make charts for a manifold, does that imply that there is any kind of metric on the manifold? ...
    (sci.math)