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@shipwright.com> Subject: Re: Questions of size... Date: Mon, 11 Dec 2000 19:16:57 -0500 To: Some People, Privately --- begin forwarded text