Krzysztof Galicki

A generalization of the momentum mapping construction for quaternionic Kåhler manifolds

We present a method of reduction of any quaternionic Kähler manifold with isometries to another quaternionic Kähler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kähler case. We compare our results with the known construction for Kähler and hyperKähler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kähler and not symmetric.

ON SASAKIAN–EINSTEIN GEOMETRY

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds and show that has the structure of a commutative associative topological monoid. The set of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in is not closed under this multiplication; however, the join of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

On Eta-Einstein Sasakian Geometry

A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold . In the case when the transverse space is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

A note on toric contact geometry

Abstract After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective torus action whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant’s theorem due to Lerman and Tolman [E. Lerman, S. Tolman, Trans. Am. Math. Soc. 349 (10) (1997) 4201–4230] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

Einstein manifolds and contact geometry

We show that every K-contact Einstein manifold is SasakianEinstein and discuss several corollaries of this result.

Betti numbers of 3-Sasakian manifolds

A vanishing theorem and constraints are given for the Betti numbers of compact 3-Sasakian manifolds.

Sasakian geometry, holonomy, and supersymmetry

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.Under the action of the c-map, special K¨ahler manifolds are mapped into a class of quaternion-K¨ahler spaces. We explicitly construct the corresponding Swann bundle or hyperk¨ahler cone, and determine the hyperk¨ahler potential in terms of the prepotential of the special K¨ahler geometry.We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems.An almost para-CR structure on a manifold

On Positive Sasakian Geometry

M

Sasakian geometry, homotopy spheres and positive Ricci curvature

is given by a distribution