342,679 research outputs found
Convexity package for momentum maps on contact manifolds
Let a torus T act effectively on a compact connected cooriented contact
manifold, and let Psi be the natural momentum map on the symplectization. We
prove that, if dim T > 2, the union of the origin with the image of Psi is a
convex polyhedral cone, the non-zero level sets of Psi are connected (while the
zero level set can be disconnected), and the momentum map is open as a map to
its image. This answers a question posed by Eugene Lerman, who proved similar
results when the zero level set is empty. We also analyze examples with dim T
<= 2.Comment: 39 pages. Contains small corrections and a small simplification of
the argument. To appear in Algebraic and Geometric Topology
Equivariant cohomology and analytic descriptions of ring isomorphisms
In this paper we consider a class of connected closed -manifolds with a
non-empty finite fixed point set, each of which is totally non-homologous
to zero in (or -equivariantly formal), where . With the
help of the equivariant index, we give an explicit description of the
equivariant cohomology of such a -manifold in terms of algebra, so that we
can obtain analytic descriptions of ring isomorphisms among equivariant
cohomology rings of such -manifolds, and a necessary and sufficient
condition that the equivariant cohomology rings of such two -manifolds are
isomorphic. This also leads us to analyze how many there are equivariant
cohomology rings up to isomorphism for such -manifolds in 2- and
3-dimensional cases.Comment: 20 pages, updated version with two references adde
A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems
We study the C*-closure A of the algebra of all operators of order and class
zero in Boutet de Monvel's calculus on a compact connected manifold X with
non-empty boundary. We find short exact sequences in K-theory
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K
denotes the compact ideal and T*X' the cotangent bundle of the interior of X.
Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we
show that the Fredholm index of an elliptic element in A is given as the
composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X'))
defined above. This relation was first established by Boutet de Monvel by
different methods.Comment: Title slightly changed. Accepted for publication in Journal fuer die
reine und angewandte Mathemati
C*-Structure and K-Theory of Boutet de Monvel's Algebra
We consider the norm closure of the algebra of all operators of order and
class zero in Boutet de Monvel's calculus on a manifold with boundary .
We first describe the image and the kernel of the continuous extension of the
boundary principal symbol to . If the is connected and is not empty,
we then show that the K-groups of are topologically determined. In case the
manifold, its boundary and the tangent space of the interior have torsion-free
K-theory, we prove that is isomorphic to the direct sum of
and , for i=0,1, with denoting the compact
ideal and the tangent bundle of the interior of . Using Boutet de
Monvel's index theorem, we also prove this result for i=1 without assuming the
torsion-free hypothesis. We also give a composition sequence for .Comment: Final version, to appear in J. Reine Angew. Math. Improved
K-theoretic result
Existence and structure of symmetric Beltrami flows on compact -manifolds
We show that for almost every given symmetry transformation of a Riemannian
manifold there exists an eigenvector field of the curl operator, corresponding
to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a
smooth, compact, oriented Riemannian -manifold with (possibly
empty) boundary and a smooth flow of isometries we show that, if has non-empty boundary or if the
infinitesimal generator is not purely harmonic, there is a smooth vector field
, tangent to the boundary, which is an eigenfield of curl and satisfies
, i.e. is invariant under the pushforward of the symmetry
transformation. We then proceed to show that if the quantities involved are
real analytic and has non-empty boundary, then Arnold's structure
theorem applies to all eigenfields of curl, which obey a symmetry and
appropriate boundary conditions. More generally we show that the structure
theorem applies to all real analytic vector fields of non-vanishing helicity
which obey some nontrivial symmetry. A byproduct of our proof is a
characterisation of the flows of real analytic Killing fields on compact,
connected, orientable -manifolds with and without boundary.Comment: 23 page
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