On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

On the holonomy of connections with skew-symmetric torsion

We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits ²-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.Mathematics Subject Classification (2000):53 C 25, 81 T 30We thank Andrzej Trautman for drawing our attention to these papers by Cartan – see [27].     

On holonomy of Weyl connections in Lorentzian signature

Holonomy algebras of Weyl connections in Lorentzian signature are classified. In particular, examples of Weyl connections with all possible holonomy algebras are constructed.     

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.     

On the finsler space admitting a holonomy group

Summary The ideas of holonomy group fixing an m-dimensional plane in a Finsler space were given by one of the present authors[1]. In that paper the deformation properties of the space admitting such holonomy group were of main consideration and, indeed, the decomposition characteristics of the space were not touched upon. In the present paper we consider the decomposition of the space due to the existence of holonomy group. The geometry is constructed on the decomposed metric of the space. The decomposition properties of various entities such as the connection parameters, the covariant derivatives, the curvature tensors, and the projective curvature tensors have been studied. In all there are six articles in the paper. The first of these is introductory. The next three articles are dealt with the Cartan's approach to Finsler space whereas the fifth one is dealt with Berwald's approach. The last article is devoted to the theory of decomposition in the projective curvature tensors. Entrata in Redazione il 18 ottobre 1969.     

Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham. Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, 1090 Vienna, Austria Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany     

On the holonomy group associated with analytic continuations of solutions for integrable systems

Necessary conditions for complex Hamiltonian systems to be integrable are considered in connection with holonomy representations of the Riemann surfaces of solutions. They are concerned with analytic continuations of solutions near those satisfying some non-resonance condition. We prove that if the system is integrable, there exists a system of local coordinates in which all Poincaré maps associated with loops on the surfaces are solved explicitly.Dedicated to Professor Kenichi Shiraiwa     

A local analytic splitting of the holonomy map on flat connections

Supported by the National Science Foundation     

Transitive holonomy group and rigidity in nonnegative curvature

In this note, we examine the relationship between the twisting of a vector bundle over a manifold M and the action of the holonomy group of a Riemannian connection on . For example, if there is a holonomy group which does not act transitively on each fiber of the corresponding unit sphere bundle, then for any , the pullback of admits a nowhere-zero cross section. These facts are then used to derive a rigidity result for complete metrics of nonnegative sectional curvature on noncompact manifolds. Received July 27, 1999; in final form November 28, 1999 / Published online February 5, 2001     

On the structure of cross connections and Galois connections

Journal of Applied Mathematics and Computing - We consider partially ordered set as passing from the set of ideals to the set of filters in Cartesian product of partially ordered sets. Lawson...     

The holonomy group of locally projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of the circle.     

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.

     

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.     

On the coincidence of group connections induced by an intrinsic composite equipment of a distribution

Mathematical Notes - In a multidimensional projective space, a distribution of planes is considered. Under the assumption that there is a relative invariant scoped by a subobject of a fundamental...     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.