Searcharxiv⌕ Search

arXiv subjects

Ilka Agricola

Publications and source records attributed to Ilka Agricola.

At least 19 recordsLinked to original sources
arXiv2026

The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this result to a large family of canonical variations of normal homogeneous metrics, thus allowing for the first time to study how the spectrum depends on the deformation parameters of the metric. As an application, we provide a formula for the full spectrum of compact positive homogeneous $3$-$(α,δ)$-Sasaki manifolds (a family of metrics which includes, in particular, all homogeneous $3$-Sasaki manifolds). The second part of the paper is devoted to the detailed computation and investigation of the spectrum of this family of metrics on the Aloff-Wallach manifold $W^{1,1}=SU(3)/S^{1}$; in particular, we provide a documented Python script that allows the explicit computation in any desired range. We recover Urakawa's eigenvalue computation for the $SU(3)$-normal homogeneous metric on $W^{1,1}$ as a limiting case and cover all the positively curved $SU(3)\times SO(3)$-normal homogeneous realizations discovered by Wilking. By doing so, we complete Urakawa's list of the first eigenvalue on compact, simply conntected, normal homogeneous spaces with positive sectional curvature.

math.DG↗
arXiv2026

Explicit Laplace Spectra of Homogeneous Principal Bundles

We present a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. For this setting, we introduce a multi-parameter family of metric deformations called generalized canonical variations. Building upon the geometric realization of such fibrations as naturally reductive spaces, we establish a simplified spectral branching criterion. We apply this method to derive the full spectra (yielding all eigenvalues and multiplicities) for several prominent geometric families. Specifically, we compute the full spectra for the entire classical series of homogeneous 3-$(α,δ)$-Sasaki manifolds (Types A, B, C, and D) and for real and complex Stiefel manifolds over Grassmannians. These explicit formulas provide the analytical data to investigate related problems in geometric analysis. As an application, we classify the scalar stability of these spaces under Perelman's $ν$-entropy and, for the 3-$(α,δ)$-Sasaki manifolds, determine the exact thresholds for Yamabe bifurcations.

math.DG↗
arXiv2026

A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms

In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^gΦ$ of the fundamental $2$-form $Φ$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_XΦ)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$.

math.DG↗
arXiv2025

Fraudulent Publishing in the Mathematical Sciences

This report is the first of two publications of a joint Working Group of the International Mathematical Union (IMU) and the International Council of Industrial and Applied Mathematics (ICIAM). In it, we shall analyze the current state of publishing in the mathematical sciences and explain the resulting problems. Our second publication will offer concrete recommendations, guidelines, and best practices for researchers, policymakers, and evaluators of mathematical research. It will explain how to detect and counteract attempts to game bibliometric measures, empowering the community to reclaim control over research evaluation and drive necessary change.

math.HO↗
arXiv2023

$\mathcal{H}$-Killing Spinors and Spinorial Duality for Homogeneous 3-$(α,δ)$-Sasaki Manifolds

We show that $3$-$(α,δ)$-Sasaki manifolds admit solutions of a certain new spinorial field equation (the $\mathcal{H}$-Killing equation) generalizing the well-known Killing spinors on $3$-Sasakian manifolds. These $\mathcal{H}$-Killing spinors have more desirable geometric properties than the spinors obtained by simply deforming a $3$-Sasakian metric; in particular we obtain a one-to-one correspondence between $\mathcal{H}$-Killing spinors on dual pairs of homogeneous $3$-$(α,δ)$-Sasaki spaces. Finally, we show that $\mathcal{H}$-Killing spinors generalize certain special spinors in dimension $7$ previously constructed by Agricola-Friedrich and Agricola-Dileo using $\text{G}_2$-geometry.

math.DG↗
arXiv2023

Invariant Spinors on Homogeneous Spheres

Using the characterization of the spin representation in terms of exterior forms, we give a complete classification of invariant spinors on the nine homogeneous realizations of the sphere $S^n$. In each of the cases we determine the dimension of the space of such spinors, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues.

math.DG↗
arXiv2022

Curvature Properties of 3-$(α,δ)$-Sasaki Manifolds

We investigate curvature properties of 3-$(α,δ)$-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to $α= δ= 1$) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to $δ= 0$, $αδ> 0$ or $αδ< 0$. We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the space of 2-forms. We describe their spectrum, find distinguished eigenforms, and study the conditions of strongly definite curvature in the sense of Thorpe.

math.DG↗
arXiv2022

Einstein manifolds with skew torsion

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar curvature, it is necessarily compact and it has finite first fundamental group. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein-Sasaki manifold and any 7-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.

math.DG↗
arXiv2021

Homogeneous non-degenerate $3$-$(α,δ)$-Sasaki manifolds and submersions over quaternionic Kähler spaces

We show that every $3$-$(α,δ)$-Sasaki manifold of dimension $4n + 3$ admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature $16n(n+2)αδ$. In the non-degenerate case ($δ\neq 0$) we describe all homogeneous $3$-$(α,δ)$-Sasaki manifolds fibering over symmetric Wolf spaces (case $αδ> 0$) and over their the noncompact dual symmetric spaces (case $αδ< 0$). If $αδ> 0$, this yields a complete classification of homogeneous $3$-$(α,δ)$-Sasaki manifolds; for $αδ< 0$, we provide a general construction of homogeneous $3$-$(α,δ)$-Sasaki manifolds fibering over nonsymmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being $19$.

math.DG↗
arXiv2018

Generalizations of 3-Sasakian manifolds and skew torsion

In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-$(α,δ)$-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result by Kashiwada. We study their behaviour under a new class of deformations, called $\mathcal{H}$-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-$(α,δ)$-Sasaki manifold is Einstein either if $α=δ$ (the 3-$α$-Sasaki case) or if $δ=(2n+3)α$, where $\dim M=4n+3$. The second part is actually devoted to finding these adapted connections. We start with a very general notion of $φ$-compatible connections, where $φ$ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-$(α,δ)$-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the $\nabla$-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated $G_2$-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

math.DG↗
arXiv2017

On the history of the Hopf problem

This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on $S^6$?" This unsolved mathematical question was the subject of the Conference "MAM 1 $-$ (Non-)Existence of Complex Structures on $S^6$", which took place at Philipps-Universität Marburg, Germany, between March 27th and March 30th, 2017.

math.HO↗
arXiv2017

$S^6$ and the geometry of nearly Kähler $6$-manifolds

We review results on and around the almost complex structure on $S^6$, both from a classical and a modern point of view. These notes have been prepared for the Workshop "(Non)-existence of complex structures on $S^6$" (\emph{Erste Marburger Arbeitsgemeinschaft Mathematik -- MAM-1}), held in Marburg in March 2017.

math.DG↗
arXiv2016

Tangent Lie groups are Riemannian naturally reductive spaces

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $TS^7$, which is in many senses almost a tangent Lie group.

math.DG↗
arXiv2015

Manifolds with vectorial torsion

The present note deals with the properties of metric connections $\nabla$ with vectorial torsion $V$ on semi-Riemannian manifolds $(M^n,g)$. We show that the $\nabla$-curvature is symmetric if and only if $V^{\flat}$ is closed, and that $V^\perp$ then defines an $(n-1)$-dimensional integrable distribution on $M^n$. If the vector field $V$ is exact, we show that the $V$-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature---for example, a $V$-Ricci-flat connection with vectorial torsion in dimension $4$, explaining some constructions occurring in general relativity. Finally, we investigate the Dirac operator $D$ of a connection with vectorial torsion. We prove that for exact vector fields, the $V$-Dirac spectrum coincides with the spectrum of the Riemannian Dirac operator. We investigate in detail the existence of $V$-parallel spinor fields; several examples are constructed. It is known that the existence of a $V$-parallel spinor field implies $dV^\flat=0$ for $n=3$ or $n\geq 5$; for $n=4$, this is only true on compact manifolds. We prove an identity relating the $V$-Ricci curvature to the curvature in the spinor bundle. This result allows us to prove that if there exists a nontrivial $V$-parallel spinor, then $\mathrm{Ric}^V=0$ for $n\neq 4$ and $\mathrm{Ric}^V(X)=X\lrcorner dV^\flat$ for $n=4$. We conclude that the manifold is conformally equivalent either to a manifold with Riemannian parallel spinor or to a manifold whose universal cover is the product of $\mathbb{R}$ and an Einstein space of positive scalar curvature. We also prove that if $dV^\flat=0$, there are no non-trivial $\nabla$-Killing spinor fields.

math.DG↗
arXiv2015

Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.

math.DG↗
arXiv2015

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the $3$-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on the $7$-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated $G_2$ structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.

math.DG↗
arXiv2014

The classification of naturally reductive homogeneous spaces in dimensions $n \leq 6$

We present a new method for classifying naturally reductive homogeneous spaces -- i.\,e.~homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion \emph{and} curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions $3,4$, and $5$, and yields as a new result the classification in dimension $6$. In each dimension, one obtains a `hierarchy' of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a `generic' parallel torsion form defines a quasi-Sasakian structure in dimension $5$ and an almost complex structure in dimension $6$.

math.DG↗