arXiv subjects

Gregor Weingart

Publications and source records attributed to Gregor Weingart.

At least 19 recordsLinked to original sources

arXiv2025

Quaternion-Kähler manifolds with non-negative quaternionic sectional curvature

Compact Hermitian symmetric spaces are Kähler manifolds with constant scalar curvature and non-negative sectional curvature. A famous result by A. Gray states that, conversely, a compact simply connected Kähler manifold with constant scalar curvature and non-negative sectional curvature is a Hermitian symmetric space. The aim of the present article is to transpose Gray's result to the quaternion-Kähler setting. In order to achieve this, we introduce the quaternionic sectional curvature of quaternion-Kähler manifolds, we show that every Wolf space has non-negative quaternionic sectional curvature, and we prove that, conversely, every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space. The proof makes crucial use of the nearly Kähler twistor spaces of positive quaternion-Kähler manifolds.

math.DG↗

arXiv2024

On quaternionic bisectional curvature

In this article we study the concept of quaternionic bisectional curvature introduced by B. Chow and D. Yang for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space. We also show that all symmetric quaternion-Kähler manifolds different from the quaternionic projective space admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

math.DG↗

arXiv2024

A Graphical Calculus for Stable Curvature Invariants

In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperk\"ahler manifolds; based on interpreting trivalent graphs with colored edges as stably invariant polynomials on the space of algebraic curvature tensors. In this graphical calculus we describe explicitly the Pfaffian polynomials central to the Theorem of Chern-Gau{\ss}-Bonnet and the normalized moment polynomials calculating the moments of sectional curvature considered as a random variable on the Gra{\ss}mannian of planes. Eventually we illustrate the power of this graphical calculus by deriving a curvature identity for compact Einstein manifolds of dimensions greater than 2 involving the Euler characteristic, the third moment of sectional curvature and the $L^2$--norm of the covariant derivative of the curvature tensor. A model implementation of this calculus for the computer algebra system Maxima is available for download under http://www.matcuer.unam.mx/~gw/CurvGraphs.mac.

math.DG↗

arXiv2023

Stability of the Non-Symmetric Space $E_7/\mathrm{PSO}(8)$

We prove that the normal metric on the homogeneous space $E_7/\mathrm{PSO}(8)$ is stable with respect to the Einstein-Hilbert action, thereby exhibiting the first known example of a non-symmetric metric of positive scalar curvature with this property.

math.DG↗

arXiv2022

Functoriality of Principal Bundles and Connections

Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and use this characterization to establish an equivalence between the category of principal bundles and a suitably defined category of functors. From the point of view of differential geometry we detail the specialization of non-linear or Ehresmann to principal and linear connections and discuss the widely known and very useful universality of principal curvature in order to characterize the vector bundles in the image of a given association functor. This is the second version.

math.DG↗

arXiv2022

On Connections and their Curvatures

This paper presents a brief study on connections on fiber, principal and vector smooth bundles as well as some relations with their curvatures.

math.DG↗

arXiv2020

Stability of Compact Symmetric Spaces

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

math.DG↗

arXiv2020

Jacobi relations on naturally reductive homogeneous spaces

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions six and seven whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of $3$--dimensional naturally reductive spaces and have the following point in common: along every geodesic the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.

math.DG↗

arXiv2017

The Standard Laplace Operator

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator $Δ$ with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operators between geometric vector bundles. Several examples are included to highlight the conclusions of this article.

math.DG↗

arXiv2017

Sequences of Orthogonal Polynomials related to Isotropy Orbits of Symmetric Spaces

Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials all have pairwise different real roots in the interval $[\,0,1\,]$. In this article the polynomials constructed by Reiswich will be identified as special cases of Jacobi polynomials thus proving the conjecture about minimal isotropy orbits of compact symmetric spaces with Dynkin diagram of type $D_m$.

math.DG↗

arXiv2017

Moments of Sectional Curvature

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distribution of the sectional curvature of Riemannian products. Moreover we calculate the moments and the distribution of the sectional curvature for all compact symmetric spaces of rank 1 explicitly and derive a formula for the moments of general symmetric spaces. Interpolating the explicit values for the moments obtained we prove a weak version of the Hitchin-Thorpe Inequality.

math.DG↗

arXiv2017

About the Eta-Invariants of Berger Spheres

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-invariants for the family of Berger metrics on the odd dimensional spheres.

math.DG↗

arXiv2017

Riemannian and Kählerian Normal Coordinates

In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power series to arbitrary order. As a byproduct we generalize Kähler normal coordinates to the class of complex affine manifolds with (1,1)-curvature tensor. Moreover we describe the Spencer connection on the infinite order Taylor series of the Kähler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.

math.DG↗

arXiv2017

Moduli Spaces of Affine Homogeneous Spaces

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{M}(\mathfrak{gl}\,V)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces of dimension dim V. Moreover we associate a $\mathrm{Sym}V^*$-comodule to a point in $\mathfrak{M}(\mathfrak{gl}\,V\,)$ and use its Spencer cohomology in order to describes the infinitesimal deformations of this point in the true moduli space $\mathfrak{M}_\infty(\mathfrak{gl}\,V\,)$.

math.DG↗

arXiv2009

Ternutator Identities

The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras.

hep-th↗

arXiv2008

Matryoshka of Special Democratic Forms

Special p-forms are forms which have components ϕ_{μ_1...μ_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form ϕ\in Λ^p R^d is called democratic if the set of nonzero components {ϕ_{μ_1...μ_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P \geq p and D \geq d. In particular, we display a remarkable nested stucture of special forms including a U(3)-invariant 2-form in six dimensions, a G_2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form Ωin ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form.

math-ph↗

arXiv2007

Special Graphs

A special p-form is a p-form which, in some orthonormal basis {e_μ}, has components ϕ_{μ_1...μ_p} = ϕ(e_{μ_1},..., e_{μ_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.

math.DG↗

arXiv2007

The Weitzenböck Machine

In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the space of Weitzenböck formulas on the one hand and characterize Weitzenböck formulas as eigenvectors for an explicitly known matrix on the other. Both classifications allow us to find tailor--suit Weitzenböck formulas for applications like eigenvalue estimates or Betti number estimates.

math.DG↗