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”.     

Geodesics on lorentzian manifolds with quasi-convex boundary

In this paper we study existence and multiplicity results of geodesics joining two given events in Lorentzian manifolds with lack of geodesic completeness. The considered Lorentzian manifolds are not necessarily static or stationary and satisfy a condition of convexity on the boundary. work supported by M.U.R.S.T. research founds 40%–60% 1992     

Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra h ? so(r, s) a polynomial pseudo-Riemannian metric of signature (r + 2, s + 2) is constructed, the holonomy algebra of this metric contains h as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to 2 and the holonomy algebras of Riemannian and Lorentzian manifolds.     

On the full holonomy group of Lorentzian manifolds

The classification of restricted holonomy groups of \(n\) -dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.     

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \mathfrakg ì \mathfrakosp(p, q|2m){\mathfrak{g}\subset\mathfrak{osp}(p, q|2m)} of Riemannian supermanifolds under the assumption that \mathfrakg{\mathfrak{g}} is a direct sum of simple Lie superalgebras of classical type and possibly of a 1-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.     

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.     

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).      

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Topological structure of complete Riemannian manifolds with cyclic holonomy groups

Let M be a complete m-dimensional Riemannian manifold with cyclic holonomy group, let X be a closed flat manifold homotopy equivalent to M, and let L→X be a nontrivial line bundle over X whose total space is a flat manifold with cyclic holonomy group. We prove that either M is diffeomorphic to X×Rm-dimX or M is diffeomorphic to L×Rm-dimX−1.     

Grassmann manifolds of Jordan algebras

We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for . Received: 18 July 2005     

New examples of compact manifolds with holonomy Spin(7)

We find new examples of compact Spin(7)-manifolds using a construction of Joyce (J. Differ. Geom., 53:89–130, 1999; Compact manifolds with special holonomy. Oxford University Press, Oxford, 2000). The essential ingredient in Joyce’s construction is a Calabi–Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds.     

Compact flat manifolds with holonomy group

In this paper we construct a family of compact flat manifolds, for all dimensions , with holonomy group isomorphic to and first Betti number zero.