Dmitri V. Alekseevsky

Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection ∇ such that ∇J is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a ∇-parallel symplectic form !. This generalises Freeds definition of (affine) special Kahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form � : C n → T ∗ C n . Such a realisation induces a canonical ∇-parallel symplectic structure on M and any special sym- plectic manifold is locally obtained this way. Special Kahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms �. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-Kahler structure on the cotangent bundle of a special Kahler manifold.

Cones over pseudo-Riemannian manifolds and their holonomy

Abstract By a classical theorem of Gallot (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.

Polyvector Super-Poincare Algebras

A class of ℤ2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W0+W1 is the maximal solvable ideal of W0 is generated by W1 and commutes with W. Choosing W1 to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W0 consists of polyvectors, i.e.all the irreducible submodules of W0 are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.

Hermitian and Kahler submanifolds of a quaternionic Kahler manifold

We study almost Hermitian submanifolds of a quaternionic Kähler manifold (M4n, Q, g). We investigate when such submanifold is Hermitian, almost Kähler and Kähler. Some characterizations of Kähler submanifolds are given. In the second part of the paper we classify Kähler manifolds which can be realized as Kähler submanifolds of a quaternionic Kähler manifold with parallel non zero second fundamental form. These manifolds can be characterized as (locally) Hermitian symmetric spaces which admit a parallel line bundle of cubic forms and coincide with Tsukada manifolds which can be realized as Kähler submanifolds of the quaternionic projective space with parallel non zero second fundamental form.

Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo-Riemannian spin manifold (M0, g0). Extending the metric g0 to a field g of bilinear forms g(p) on TpM, p ϵ M0, the pseudo-Riemannian supergeometry of (M, g) is formulated as G-structure on M, where G is a supergroup with even part G0 ≊ Spin(k, l); (k, l) the signature of (M0, go). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field Xs on M. Our main result is that Xs is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field Xs.

Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.

POISSON STRUCTURES ON THE COTANGENT BUNDLE OF A LIE GROUP OR A PRINCIPLE BUNDLE AND THEIR REDUCTIONS

On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself), and also in terms of the left Maurer–Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on T*G and to Poisson structures by replacing the canonical momenta of the right or left actions of G on itself by arbitrary ones, followed by reduction (to G cross a Weyl‐chamber, e.g.). This method also works on principal bundles.

Quaternionic Kähler metrics associated with special Kähler manifolds

We give an explicit formula for the quaternionic Kahler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kahler. A similar explicit formula is given for the analogous (K/K) correspondence between Kahler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kahler manifold.

Two-symmetric Lorentzian manifolds

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen–Wallach spaces, they have commutative holonomy.

Lifting smooth curves over invariants for representations of compact lie groups

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.