No, I'm the one who's missing something -- it's dumb of me to spontaneously add comments I haven't thought out! Of course you are right -- there's no problem in giving a topological example of (round, but not congruent) k-spheres foliating an open set of R^(k+p), for any p >= 1. (It's a perfectly good foliation that you described, of the highest differentiability class, even -- real analytic.) It's a fascinating fact about foliations, by the way, that the 3-sphere admits many C^oo codimension-1 foliations (i.e., by surfaces), but does not admit any real analytic one. --Dan ------------ Allan wrote: << I think I must be missing something. Take a family of parallel, concentric 2-spheres whose radii span the open interval (1,2). Say they are in the abc-hyperplane. Their union is a 3-dimensional set. Now cross that set with an open 3-ball in the def-hyperplane. The resulting set is open, and 6-dimensional. I don't think any of the 2-spheres intersect any other. Doesn't that answer your question constructively, and in the affirmative? Maybe I'm missing one of the requirements for foliation. On Fri, Mar 27, 2009 at 10:51 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks to Fred for pointing out that I couldn't have meant 2-spheres in R^5 (that case is already excluded, as mentioned).
What I meant to write is 2-spheres in R^6. (In fact, even just topologically: I don't know if topological 2-spheres can foliate a non-empty open set of R^6.) . . .
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele