57 research outputs found
Riemannian Supergeometry
Motivated by a paper of Zirnbauer, we develop a theory of Riemannian
supermanifolds up to a definition of Riemannian symmetric superspaces. Various
fundamental concepts needed for the study of these spaces both from the
Riemannian and the Lie theoretical viewpoint are introduced, e.g. geodesics,
isometry groups and invariant metrics on Lie supergroups and homogeneous
superspaces.Comment: 43 pages, abridged version of author's thesi
Torus actions whose equivariant cohomology is Cohen-Macaulay
We study Cohen-Macaulay actions, a class of torus actions on manifolds,
possibly without fixed points, which generalizes and has analogous properties
as equivariantly formal actions. Their equivariant cohomology algebras are
computable in the sense that a Chang-Skjelbred Lemma, and its stronger version,
the exactness of an Atiyah-Bredon sequence, hold. The main difference is that
the fixed point set is replaced by the union of lowest dimensional orbits. We
find sufficient conditions for the Cohen-Macaulay property such as the
existence of an invariant Morse-Bott function whose critical set is the union
of lowest dimensional orbits, or open-face-acyclicity of the orbit space.
Specializing to the case of torus manifolds, i.e., 2r-dimensional orientable
compact manifolds acted on by r-dimensional tori, the latter is similar to a
result of Masuda and Panov, and the converse of the result of Bredon that
equivariantly formal torus manifolds are open-face-acyclic.Comment: 28 pages. V2 contains changes according to the referee's suggestion
Torsion in equivariant cohomology and Cohen-Macaulay G-actions
We show that the well-known fact that the equivariant cohomology of a torus
action is a torsion-free module if and only if the map induced by the inclusion
of the fixed point set is injective generalises to actions of arbitrary compact
connected Lie groups if one replaces the fixed point set by the set of points
with maximal isotropy rank. This is true essentially because the action on this
set is always equivariantly formal. In case this set is empty we show that the
induced action on the set of points with highest occuring isotropy rank is
Cohen-Macaulay. It turns out that just as equivariant formality of an action is
equivalent to equivariant formality of the action of a maximal torus, the same
holds true for equivariant injectivity and the Cohen-Macaulay property. In
addition, we find a topological criterion for equivariant injectivity in terms
of orbit spaces.Comment: 14 pages, 1 figur
On the Geometry of the Orbits of Hermann Actions
We investigate the submanifold geometry of the orbits of Hermann actions on
Riemannian symmetric spaces. After proving that the curvature and shape
operators of these orbits commute, we calculate the eigenvalues of the shape
operators in terms of the restricted roots. As applications, we get a formula
for the volumes of the orbits and a new proof of a Weyl-type integration
formula for Hermann actions.Comment: 22 page
K-cosymplectic manifolds
In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic
manifolds for which the Reeb field is Killing with respect to some Riemannian
metric. These structures generalize coK\"ahler structures, in the same way as
K-contact structures generalize Sasakian structures. In analogy to the contact
case, we distinguish between (quasi-)regular and irregular structures; in the
regular case, the K-cosymplectic manifold turns out to be a flat circle bundle
over an almost K\"ahler manifold. We investigate de Rham and basic cohomology
of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector
fields and group actions on such manifolds. The deformations of type I and II
in the contact setting have natural analogues for cosymplectic manifolds; those
of type I can be used to show that compact K-cosymplectic manifolds always
carry quasi-regular structures. We consider Hamiltonian group actions and use
the momentum map to study the equivariant cohomology of the canonical torus
action on a compact K-cosymplectic manifold, resulting in relations between the
basic cohomology of the characteristic foliation and the number of closed Reeb
orbits on an irregular K-cosymplectic manifold.Comment: 33 pages, no figures. Comments are welcome! V2 with more references
and some corrections (Proposition 3.5 in V1 deleted
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