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08.10.05 10:12  Математический семинар Глобус, заседание 13 октября 2005 г.

В четверг, 13 октября 2005 года, в 15:40 в конференц-зале НМУ, Б. Власьевский, 11 состоится доклад «Skew loops, skew branes, totally skew submanifolds and other curiosities». Лектор – Сергей Табачников (PennState University).

In the 1960s, H. Steinhaus asked whether every closed curve in space has a pair of parallel tangent lines; a curve without parallel tangents is called a skew loop. I will survey work of a number of authors that stemmed from this question (starting with B. Segre).

Here is a sample of results.

1. Skew loops have an eversion to quadratic surfaces: a closed curve on quadric in 3-space has parallel tangent lines. But every convex non-quadratic surface carries a skew loop.

2. Skew loops also have eversion to ruled developable surfaces.

3. A skew brane is a multidimensionaal analog of a skew loop: it is a codimension 2 submanifold without parallel tangent spaces. Skew branes also have an eversion to quadratic hypersurfaces.

4. There are no closed skew branes with non-zero Euler characteristic, but there exist skew tori and odd-dimensional skew spheres.

5. A submanifold in affine space is totally skew if any two tangent lines at disticts points are neither parallel nor intersect. A generic n-dimensional submanifols in 4n+1-dimensional space is totally skew. There exist totally skew n-dimensional spheres in 3n+2-space, and in 3n+1-space if n is odd.

6. Denote by N(n) the least dimension of space into which n-space can be embedded as a totally skew submanifold. Then N(n)\geq 2n+2, unless n=1,3,7. One has the following lower bounds for the numbers N(n):

(n, N(n)\geq) = (1, 3) (2, 6) (3, 7) (4, 12) (5, 13) (6, 14) (7, 15) (8, 24) (9, 25) (10, 27) (11, 28) (12, 31) (13, 36) (14, 37) (15, 38) (16, 48) (17, 49).

The first two estimates are sharp.