Noncompact Riemannian spaces with the holonomy group spin(7) and 3-Sasakian manifolds

We complete the study of the existence of Riemannian metrics with Spin(7) holonomy that smoothly resolve standard cone metrics on noncompact manifolds and orbifolds related to 7-dimensional 3-Sasakian spaces.     

Affine manifolds with diagonal holonomy

As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the changes of charts are restrictions of elements of the subgroup of Aff ( \mathbbRⁿ{\mathbb{R}^n}) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular, we study some properties of “Benoist manifolds”.     

New examples of manifolds with negative curvature

This lecture is about the global structure of Riemannian manifolds with negative sectional curvature and finite volume. A fundamental question is to find and analyze geometric conditions which ensure that Mⁿ has finitely many ends or, stronger, that Mⁿ is of finite topological type.     

On the new examples of complete noncompact Spin(7)-holonomy metrics

We construct some complete Spin(7)-holonomy Riemannian metrics on the noncompact orbifolds that are ?⁴-bundles with an arbitrary 3-Sasakian spherical fiber M. We prove the existence of the smooth metrics for M = S ⁷ and M = SU(3)/U(1) which were found earlier only numerically.     

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kähler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

Properties of manifolds with skew-symmetric torsion and special holonomy

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structures, thus answering a question in Grantcharov and Poon (Comm. Math. Phys. 213 (2000) 19). We also prove that the existence of an HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure whose Bismut connection has restricted holonomy in SU(n), thus providing a counter-example to the conjecture in Gutowski et al. (Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class, math.DG/0205012, Asian J. Math., to appear). Again we prove that such a property is not stable under small deformations.     

Codazzi spinors and globally hyperbolic manifolds with special holonomy

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface for any weakly irreducible holonomy representation with parallel spinors, t.m. with a holonomy group , where is trivial or a product of groups SU(k), Sp(l), G ₂ or Spin (7).