28 research outputs found
Isoparametric and Dupin Hypersurfaces
A hypersurface Mn−1 in a real space-form Rn, Sn or Hn is isoparametric if it has constant principal curvatures. For Rn and Hn, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sn. A hypersurface Mn−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field
New examples of Willmore submanifolds in the unit sphere via isoparametric functions,II
This paper is a continuation of a paper with the same title of the last two
authors. In the first part of the present paper, we give a unified geometric
proof that both focal submanifolds of every isoparametric hypersurface in
spheres with four distinct principal curvatures are Willmore. In the second
part, we completely determine which focal submanifolds are Einstein except one
case.Comment: 19 pages,to appear in Annals of Global Analysis and Geometr
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides
Cyclidic nets are introduced as discrete analogs of curvature line
parametrized surfaces and orthogonal coordinate systems. A 2-dimensional
cyclidic net is a piecewise smooth -surface built from surface patches of
Dupin cyclides, each patch being bounded by curvature lines of the supporting
cyclide. An explicit description of cyclidic nets is given and their relation
to the established discretizations of curvature line parametrized surfaces as
circular, conical and principal contact element nets is explained. We introduce
3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate
systems and investigate them in detail. Our considerations are based on the Lie
geometric description of Dupin cyclides. Explicit formulas are derived and
implemented in a computer program.Comment: 39 pages, 30 figures; Theorem 2.7 has been reformulated, as a
normalization factor in formula (2.4) was missing. The corresponding
formulations have been adjusted and a few typos have been correcte
Access to Primary Care and Visits to Emergency Departments in England: A Cross-Sectional, Population-Based Study
10.1371/journal.pone.0066699PLoS ONE86e6669