Lorenz Schwachhoefer

Technische Universität Dortmund | TUD

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W*-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on pa...

Inspired by Kirillov's theory of coadjoint orbits, we develop a structure theory for finite dimensional Jordan algebras. Given a Jordan algebra ${\mathcal{J}}$, we define a generalized distribution $\mathcal{H}^{{\mathcal{J}}}$ on its dual space ${\mathcal{J}}^\star$ which is canonically determined by the Jordan product in ${\mathcal{J}}$, is invar...

A geometrical formulation of estimation theory for finite-dimensional C*-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented.

A geometrical formulation of estimation theory for finite-dimensional $C^{\star}$-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is p...

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures o...

The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved tha...

The Jordan product on the self-adjoint part of a finite-dimensional $C^{*}$-algebra $\mathscr{A}$ is shown to give rise to Riemannian metric tensors on suitable manifolds of states on $\mathscr{A}$, and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitely computed. In p...

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Frölicher–Nijenhuis bracket. This bracket enables us to define the Frölicher–Nijenhuis cohomologies which are analogues of the and the Dolbeault cohomologies in Kähler geometry, and assigns an -algebra to each associative submanifold. We provide...

In this paper we introduce the notion of Poincar\'e DGCAs of Hodge type, which is a subclass of Poincar\'e DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincar\'e DGCA of Hodge type. Using these concepts, we investigate the equivalence c...

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Fr\"olicher-Nijenhuis bracket. This bracket enables us to define the Fr\"olicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in K\"ahler geometry, and assigns an $L_\infty$-algebra to each associative s...

In this paper, we show that a parallel differential form ψ of even degree on a Riemannian manifold allows to define a natural differential both on Ω(M) and Ω(M,TM), defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigat...

We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space which does not assume that all measures of the model have the same null sets. This is given by a differentiable map from the parameter manifold M into the set of finite measures or probability measures on , respectively, which is dif...

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi\vert_N\in \Omega^*(N, TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compac...

A classical result of Chentsov states that – up to constant multiples – the only 2-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant 3-tensor field is the Amari–Chentsov tensor. We generalize this result for arbitrary degree n, showing that any family of n-tensors whi...

We introduce the notions of essential tangent space and reduced Fisher metric and extend the classical Cramér-Rao inequality to 2-integrable (possibly singular) statistical models for general \(\varphi \)-estimators, where \(\varphi \) is a V-valued feature function and V is a topological vector space. We show the existence of a \(\varphi \)-effici...

Complexity measures can be geometrically built by using the information distance (Kullback–Leibler divergence) from families with restricted statistical dependencies. The Pythagorean geometry developed in Chaps. 2 and 4 allows us to iterate such constructions, by going to simpler and simpler families and taking—possibly weighted—sums, thereby syste...

This chapter investigates probability distributions on a finite sample space and takes advantage of the more elementary nature of this setting. There are two complementary ways to view a probability distribution. One consists in viewing it as (positive) measure with total mass 1. The other considers it as an equivalence class of such measures, dete...

Classical results of Chentsov and Campbell state that -- up to constant multiples -- the only $2$-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant $3$-tensor field is the Amari-Chentsov tensor. We generalize this result for arbitrary degree $n$, showing that any fami...

We introduce the notion of the essential tangent bundle of a parametrized measure model and the notion of reduced Fisher metric on a (possibly singular) 2-integrable measure model. Using these notions and a new characterization of $k$-integrable parametrized measure models, we extend the Cram\'er-Rao inequality to $2$-integrable (possibly singular)...

In this paper we show that a parallel differential form $\Psi$ of even degree on a Riemannian manifold allows to define a natural differential both on $\Omega^\ast(M)$ and $\Omega^\ast(M, TM)$, defined via the Fr\"olicher-Nijenhuis bracket. For instance, on a K\"ahler manifold, these operators are the complex differential and the Dolbeault differen...

In this paper we extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Fr\"olicher-Nijenhuis bracket $[J, J] ^{FN}$ to analogous characterizations of torsion-free $G_2$-structures and torsion-free ${\rm Spin}(7)$-structures. We also explain the Fern\'andez-Gray class...

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$. This is given by a diffferentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $\Omega$, respectively, which is differentiable when regarded as a map into the Banach space...

We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and Pistone-Sempi [10]. We give a complete description of n-tensor fields that are invariant under sufficient statistics. In the cases \(n= 2\) and \(n = 3\), the only such tensors are the Fis...

Lagrangian submanifolds in strictly nearly K\"ahler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly K\"ahler manifold $(M^{2n}, J, g)$ is symplectically dual to the Maslov 1-form on $L$. Using...

An affine connection is one of the basic objects of interest in differential geometry. 4 It provides a simple and invariant way of transferring information from one 5 point of a connected manifold M to another and, not surprisingly, enjoys lots of 6 applications in many branches of mathematics, physics and mechanics.

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 th...

We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.

The field of geometric variational problems is fast-moving and influential. These problems interact with many other areas of mathematics and have strong relevance to the study of integrable systems, mathematical physics and PDEs. The workshop 'Variational Problems in Differential Geometry' held in 2009 at the University of Leeds brought together in...

We construct a canonical linear resolution of acyclic 1-dimensional sheaves on P^1 x P^1 and discuss the resulting natural Poisson structure.

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, ho...

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, \Omega)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affine normal space at $\imat...

Let (V, Ω) be a symplectic vector space and let f: M ® V{\phi : M \rightarrow V} be a symplectic immersion. We show that f(M) Ì V{\phi(M) \subset V} is locally an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of Cahen etal. (J Geom Phys 59(4):409f́b-425, 2009) if and only if the second fundamental form of f{\phi} is parallel. Further...

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new...

We consider cohomogeneity one homogeneous disk bundles and address the question when these admit a nonnegatively curved† invariant metric with normal collar, that is, such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its...

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural star-quantization on a class of examples.

We apply the results from the article Cahen, Schwachh\"ofer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in $su(n,1)$. The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metrics constructions is discusse...

We show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\h$ of $\g$ always has some negative sectional curvature, unless the semi-simple part of $\h$ is an ideal of $\g$.

We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching...

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Khler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic....

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplect...

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...

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonnegative Ricci and almost nonne...

We shall discuss Riemannian metrics of fixed diameter and controlled lower curvature bound. As in [34], we give a general construction of invariant metrics on homogeneous vector bundles of cohomogeneity one, which implies, in particular, that any cohomogeneity one manifold admits invariant metrics of almost nonnegative sectional curvature. This pro...

We classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy, i.e. a connection whose holonomy is an absolutely irreducible proper subgroup of the full symplectic group. Thereby, we obtain many new explicit descriptions of manifolds with special symplectic holonomies. We also show that manifolds wit...

We show that any closed cohomogeneity one manifold supports metrics of almost nonnegative sectional curvature which are moreover invariant under the cohomogeneity one action, thereby establishing a conjecture of Grove and Ziller in the almost nonnegatively curved setting. Applications of our result include that there are infinitely many dimensions...

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 Gray's contributions in this area on the one han...

The real form Spin(6,H) in End(R^{32}) of Spin(12,C) in End(C^{32}) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41). Therefore, it also occurs as an exotic holonomy and the associated supermanifold M_g admits a SUSY-invariant polynomial. This real form has been erroneously omitted in our paper. Also, the two real...

The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poiss...

It is proved that the Lie groups and represented in ℝ56 and the Lie group represented in ℝ112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension...

In 1955, Berger [4] gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. While this list was stated to be complete in the case of metric connections, the situation in the general case remained unclear. The (non-metric) representations which are missing from this list are called ex...

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 l...

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...

It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in $\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connecti...

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 l...

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 co...

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 [Bry] that Berger's list is incomplete. Connections whose holonomy is not contained on Berger's list are called exotic. We investigate a certain 4-dimensional exotic hol...

We classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy, i.e. a connection whose holonomy is an absolutely irreducible proper subgroup of the full symplectic group. Thereby, we obtain many new explicit descriptions of manifolds with special symplectic holonomies. We also show that manifolds wit...

On a given symplectic manifold, there are many symplectic connections, i.e. torsion free connections w.r.t. which the symplectic form is parallel. We call such a connection special if it is either the Levi-Civita connection of a Bochner-Kahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connect...

The search for manifolds of nonnegative curvature1 is one of the classical problems in Riemannian geometry. While general obstructions are scarce, there are relatively few general classes of examples and construction methods. Hence, it is unclear how large one should expect the class of closed manifolds admitting a nonnegatively curved metric to be...