Questions of size...

[Somebody]

Bob,

The distinction between geometry, topology, and, presumably, homology, begs
the question.

You, following Huber, have used the word geodesic to refer to connections
of minimal cost. In the economic manifold (surface) this is presumably the
only important metric (local distance function).  The analogy is
mathematically precise in all respects, and therefore correct.

I'd have flogged you into submission long before this if it were not so.

Geodesics, or more properly, geodesic paths, are locally defined.  There is
not necessarily a geodesic between two specific points.  In a
differentiable manifold with a sufficiently smooth metric, there are
geodesic paths in every direction through every point, however.  Whether
there is one of those paths going to some other given point is a question
of connectivity and other topological issues.  Two separate spheres are a
single differential manifold.  No great circle path -- the geodesics of
each surface -- connects any point on one sphere with any point of the
other.

Unfortunately, geodesics may also be the longest paths between two points.
Just go the wrong way on the great circle determined by two ends of the
Mass Ave bridge.  It's a path of stationary length: slight variations in
the path make hardly any difference in its length.  Unfortunately its
length is maximum rather than minimum.

By the way, have I mentioned that I HATE it when I agree with Tim?


<Somebody's .sig>

------------------------
  From: "R. A. Hettinga" <rah at shipwright.com>
  Subject: Re: Questions of size...
  Date: Mon, 11 Dec 2000 19:16:57 -0500
  To: Some People, Privately



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