1,645 research outputs found
On the Maslov class rigidity for coisotropic submanifolds
We define the Maslov index of a loop tangent to the characteristic foliation
of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the
group of linear symplectic transformations, incorporating the "rotation" of the
tangent space of the leaf -- this is the standard Lagrangian counterpart -- and
the holonomy of the characteristic foliation. Furthermore, we show that, with
this definition, the Maslov class rigidity extends to the class of the
so-called stable coisotropic submanifolds including Lagrangian tori and stable
hypersurfaces.Comment: 18 pages; v2 minor corrections, references update
Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case
The paper is centered around a new proof of the infinitesimal rigidity of
smooth closed surfaces with everywhere positive Gauss curvature. We use a
reformulation that replaces deformation of an embedding by deformation of the
metric inside the body bounded by the surface. The proof is obtained by
studying derivatives of the Hilbert-Einstein functional with boundary term.
This approach is in a sense dual to proving the Gauss infinitesimal rigidity,
that is rigidity with respect to the Gauss curvature parametrized by the Gauss
map, by studying derivatives of the volume bounded by the surface. We recall
that Blaschke's classical proof of the infinitesimal rigidity is also related
to the Gauss infinitesimal rigidity, but in a different way: while Blaschke
uses Gauss rigidity of the same surface, we use the Gauss rigidity of the polar
dual. In the spherical and in the hyperbolic-de Sitter space, there is a
perfect duality between the Hilbert-Einstein functional and the volume, as well
as between both kinds of rigidity. We also indicate directions for future
research, including the infinitesimal rigidity of convex cores of hyperbolic
3--manifolds.Comment: 60 page
The sigma orientation for analytic circle-equivariant elliptic cohomology
We construct a canonical Thom isomorphism in Grojnowski's equivariant
elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with
vanishing Borel-equivariant second Chern class, which is natural under
pull-back of vector bundles and exponential under Whitney sum. It extends in
the complex-analytic case the non-equivariant sigma orientation of Hopkins,
Strickland, and the author. The construction relates the sigma orientation to
the representation theory of loop groups and Looijenga's weighted projective
space, and sheds light even on the non-equivariant case. Rigidity theorems of
Witten-Bott-Taubes including generalizations by Kefeng Liu follow.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper3.abs.htm
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