How the Curvature Generates the Holonomy of a Connection in an Arbitrary Fibre Bundle

For an arbitrary fibre bundle with a connection, the holonomy group of which is a Lie transformation group, it is shown how the parallel displacement along a null-homotopic loop can be obtained from the curvature by integration. The result also sheds some new light on the situation for vector bundles and principal fibre bundles. The Theorem of Ambrose–Singer is derived as a corollary in our general setting. The curvature of the connection is interpreted as a differential 2-form with values in the holonomy algebra bundle, the elements of which are special vector fields on the fibres of the given bundle. Received: May 16, 2006; Revised: July 30, 2006; Accepted: August 2, 2006     

Extrinsic homogeneity of parallel submanifolds II

We discuss the question whether a (complete) parallel submanifold M of a Riemannian symmetric space N is an (extrinsically) homogeneous submanifold, i.e. whether there exists a subgroup of the isometries of N which acts transitively on M. In a previous paper, we have discussed this question in case the universal covering space of M is irreducible. It is the subject of this paper to generalize this result to the case when the universal covering space of M has no Euclidian factor.     

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.     

A theorem on the volume growth in non-positive curved manifolds

We prove a theorem about the volume growth in non-positive curved manifolds under a condition on the Ricci curvature.     

Mirror duality in a Joyce manifold

Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G₂ manifold, and described a process to obtain them. Here we apply this process to a compact G₂ manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.     

Representations admitting two pairs of supplementary invariant spaces

We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector-space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of supplementary invariant spaces, or one pair and a reflexive form. We show that such a representation is a direct sum of three canonical sub-representations which we characterize. We then focus on representations of Berger algebras with the same property.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.     

Variational sets: Calculus and applications to nonsmooth vector optimization

We develop elements of calculus of variational sets for set-valued mappings, which were recently introduced in Khanh and Tuan (2008) [1] and [2] to replace generalized derivatives in establishing optimality conditions in nonsmooth optimization. Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established. Direct applications in stability and optimality conditions for various vector optimization problems are provided.     

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     

Über Ableitungsbegriffe auf fastkomplexen Mannigfaltigkeiten

Ohne Zusammenfassung

---

This article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28 – September 3, 1975.     

Conformally parallel G 2 structures on a class of solvmanifolds

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G₂-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G₂ structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G₂. In the process we also find a new metric with exceptional holonomy. Received: 20 September     

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.     

On the concept of generalized order optimality

The main results of this paper include a detailed analysis of the notion of generalized order optimality and some sufficient conditions for a point satisfying the necessary optimality condition of Mordukhovich (2006) [1] and [2] for being a generalized order solution of the optimization problem under consideration.     

Caractérisation des feuilletages holomorphes singuliers, contenus dans des feuilletages Levi flat, sur les surfaces complexes compactes

Let be a holomorphic foliation with reduced singularities on a complex surface M and a real analytic codimension one foliation on M whose leaves contain the ones of . We show that a Levi flat group of diffeomorphisms of is resoluble and holomorphically conjugate to his normal form. We deduce, in one hand, that each singularity of  is conjugate to his normal form. In the other hand at each singularity m of , where is not defined, up a conjugacy, by the one form ω=xdy+ydx, one of the local invariant curves of , with non obvious holonomy, is contained in the set of singularities of . Moreover if M is a compact Stein variety we show, under some generic conditions, that has a 1-Liouvillian first integrating factor.