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.) --Dan I wrote: << On 3/28/09, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. For completeness I'll add that the above readily implies that there do exist non-empty open sets in R^(2k+1+p), for any p >= 0, that are foliated by congruent round k-spheres for k = 0,1,3,7. And conversely, for any k not in {0,1,3,7}, there does not exist a non-empty open set in R^(2k+1-p), for p >= 0 and 2k+1-p >= 0, that is foliated by congruent round k-spheres.
But this leaves unresolved infinitely many cases of k-spheres in n-space. First open case: Can congruent round 2-spheres foliate a non-empty open set in R^5 ??? ^^^
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele