arXiv subjects

Leander Stecker

Publications and source records attributed to Leander Stecker.

7 recordsLinked to original sources

arXiv2026

Reducible Holonomy in Closed Torsion Geometries

The purpose of this note is to show that a connection with closed skewsymmetric torsion and reducible holonomy admits a locally defined Riemannian submersion together with a projected geometry on the base. We reframe known submersion results for non-K\"ahler Bismut Hermite Einstein manifolds and sHKT structures in this context. For homogeneous SKT structures on semi-simple Lie groups we obtain the holonomy decomposition leading to holomorphic submersions over generalized flag manifolds.

math.DG↗

arXiv2024

The heterotic G$_2$ system with reducible characteristic holonomy

We construct solutions to the heterotic G$_2$ system on almost contact metric manifolds with reduced characteristic holonomy. We focus on $3$-$(\alpha,\delta)$-Sasaki manifolds and $(\alpha,\delta)$-Sasaki manifolds, the latter being a convenient reformulation of spin $\eta$-Einstein $\alpha$-Sasaki manifolds. Investigating a $1$-parameter family of G$_2$-connections on the tangent bundle, we obtain several approximate solutions as well as one new class of exact solutions on degenerate $3$-$(\alpha,\delta)$-Sasaki manifolds.

math.DG↗

arXiv2022

Canonical Submersions in Nearly K\"ahler Geometry

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application we show that parallel 3-$(\alpha,\delta)$-Sasaki manifolds admit 1-dimensional submersions onto nearly K\"ahler orbifolds. As a secondary application we reprove that a certain class of nearly K\"ahler manifolds submerges onto quaternionic K\"ahler manifolds. This new proof gives an direct expression for the quaternionic structure on the base.

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

On Degenerate $3$-$(α,δ)$-Sasakian Manifolds

We propose a new method to construct degenerate $3$-$(α,δ)$-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate $3$-$(α, δ)$-Sasakian manifolds and prove that no non-trivial compact examples exist as well as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

math.DG↗

arXiv2021

Revisiting the Classification of Homogeneous 3-Sasakian and Quaternionic Kähler Manifolds

We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann. In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss why the real projective spaces are the only non-simply connected homogeneous 3-Sasakian manifolds and derive the famous classification of homogeneous positive quaternionic Kähler manifolds due to Wolf and Alekseevskii from our results.

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↗