Lorenz J. Schwachhöfer

Information geometry and sufficient statistics

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari–Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari–Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.

On the Existence of Infinite Series of Exotic Holonomies

In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Bergers classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with these new holonomies. We also deduce some striking facts about their global behaviour.

Exotic holonomy on moduli spaces of rational curves

Abstract Bryant [3] proved the existence of torsion free connections with exotic holonomy, i.e., with holonomy that does not occur on the classical list of Berger [1]. These connections occur on moduli spaces Y of rational contact curves in a contact threefold W. Therefore, they are naturally contained in the moduli space Z of all rational curves in W. We construct a connection on Z whose restriction to Y is torsion free. However, the connection on Z has torsion unless both Y and Z are flat. This answers a question of Bryant as to whether the GL (2, C ) × SL (2, C )-structures which arise from such a moduli space Z always admit a torsion free connection in the negative. We also show the existence of a new exotic holonomy which is a certain six-dimensional representation of SL (2, C ) × SL (2, C ). We show that every regular H 3 -connection (cf. [3]) is the restriction of a unique connection with this holonomy.

Riemannian, Symplectic and Weak Holonomy

Much of the early work of Alfred Gray was concerned with the investigation of Riemannian manifolds with special holonomy, one of the most vivid fields of Riemannian geometry in the 1960s and the following decades. It is the purpose of the present article to give a brief summary and an appreciation of Grays contributions in this area on the one hand, and on the other hand to describe some of the more recent developments in the theory of non-Riemannian or,more specifically, symplectic holonomy groups. Namely, we show that the Merkulov twistor space of a connection on a symplectic manifold M whose holonomy group is irreducible and properly contained in Sp(V) consists of maximal totally geodesic Lagrangian submanifolds of M.

Connections with exotic holonomy

CONNECTIONS WITH EXOTIC HOLONOMY Lorenz Johannes Schwachhofer Wolfgang Ziller (Supervisor) In 1955, Berger [Ber] partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold. However, it was shown by Bryant [Br2] that Berger’s list is incomplete. Connections whose holonomy is not contained on Berger’s list are called exotic. We investigate examples of a certain 4-dimensional exotic holonomy representation of Sl(2,R). We show that connections with this holonomy are never complete, give explicit descriptions of these connections on an open dense set and compute their group of symmetry. Finally, we give strong restrictions for their existence on compact manifolds.

Construction of Ricci-type connections by reduction and induction

Given the Euclidean space ℝ2n+2 endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension 2n (n ≥ 2) endowed with a symplectic connection of Ricci type is locally given by a local version of such a reduction.

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 Bergers classical list of irreducible holonomy representations. The holonomy in Bryants examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H3-connections (G3-connections resp.).In this paper, we give a complete classification of homogeneous G3-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G3-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G3-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 H3-connection.

Hypercomplex limits of pluricomplex structures and the Euclidean limit of hyperbolic monopoles

We discuss the Euclidean limit of hyperbolic

{\rm E}^{(a)}_7

SU(2)