168 research outputs found
On skew loops, skew branes and quadratic hypersurfaces
A skew brane is an immersed codimension 2 submanifold in affine space, free
from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew
brane cannot lie on a quadratic hypersurface. We also prove that there are no
skew loops on embedded ruled developable discs in 3-space. The paper extends
recent work by M. Ghomi and B. Solomon.Comment: 13 pages, 2 figure
Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem
We interpret magnetic billiards as Finsler ones and describe an analog of the
string construction for magnetic billiards. Finsler billiards for which the law
"angle of incidence equals angle of reflection" are described. We characterize
the Finsler metrics in the plane whose geodesics are circles of a fixed radius.
This is a magnetic analog of Hilbert's fourth problem asking to describe the
Finsler metrics whose geodesics are straight lines.Comment: 27 pages, 6 figure
On three-periodic trajectories of multi-dimensional dual billiards
We consider the dual billiard map with respect to a smooth strictly convex
closed hypersurface in linear 2m-dimensional symplectic space and prove that it
has at least 2m distinct 3-periodic orbits.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-33.abs.htm
The Six Circles Theorem revisited
The Six Circles Theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell
concerns chains of circles inscribed into a triangle: the first circle is
inscribed in the first angle, the second circle is inscribed in the second
angle and tangent to the first circle, the third circle is inscribed in the
third angle and tangent to the second circle, and so on, cyclically. The
theorem asserts that if all the circles touch the sides of the triangle, and
not their extensions, then the chain is 6-periodic. We show that, in general,
the chain is eventually 6-periodic but may have an arbitrarily long pre-period.Comment: edited for better expositio
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