arXiv subjects

Ioannis Chrysikos

Publications and source records attributed to Ioannis Chrysikos.

At least 19 recordsLinked to original sources

arXiv2026

Adapted connections with skew-torsion on metric $f$-manifolds

We provide a natural higher-dimensional generalization of the adapted connections with skew-torsion on almost Hermitian and almost contact metric manifolds presented in \cite{FrIv}. We prove that a metric $f$-manifold $(M^{2n+s}, ϕ, ξ_i, η_j, g)$ with commuting characteristic vector fields admits a metric connection $\nabla$ with skew-torsion $T$ preserving the structure if and only if each Reeb vector field $ξ_i$ is Killing and the associated Nijenhuis tensor is totally skew-symmetric. This connection is uniquely determined by its torsion 3-form $T$, for which we derive its explicit formula. We further establish necessary and sufficient conditions for contact metric $f$-manifolds to admit such a connection. To this end, for $s\geq 2$ we construct a broad new class of geometries with parallel skew-torsion in terms of $\mathcal{S}$-manifolds, i.e., higher-dimensional analogues of Sasakian manifolds. We illustrate our results by presenting examples based on low-dimensional Lie groups.

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arXiv2026

The canonical submersion of $\mathcal{S}$-manifolds and transverse Kähler-Einstein structures

This paper is devoted to the study of the holonomy properties of $(2n+s)$-dimensional $\mathcal{S}$-manifolds equipped with their characteristic connection. These structures generalize Sasakian geometry to higher CR-codimensions and, when viewed as geometries with parallel skew-torsion, share many holonomy features with the Sasakian case. We show that $\mathcal{S}$-manifolds of arbitrary CR-codimension $s\geq 1$ provide examples of geometries with parallel skew-torsion whose holonomy is reducible, indecomposable, and of special type. We also deduce that any $\mathcal{S}$-manifold admits a locally defined Riemannian submersion over a Kähler manifold. We describe the corresponding curvature relations and establish a bijective correspondence between the Kähler-Einstein condition on the base space and a generalized $η$-Einstein condition on the total space. As every $\mathcal{S}$-manifold comes with a characteristic foliation whose transverse geometry is Kähler, it is natural to relate the $η$-Einstein condition to the transverse metric, leading to a bijection between $η$-Einstein $\mathcal{S}$-manifolds and transverse Kähler-Einstein metrics, extending the classical Sasakian correspondence to arbitrary CR-codimensions. As an application to Ricci-flat metric connections with parallel skew-torsion, we prove that an $\mathcal{S}$-manifold is ${\rm Ric}^{\nabla}$-flat if and only if it is transverse Kähler-Einstein with Einstein constant $λ=4s$. An illustration of this characterization is presented by a Sasakian example.

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arXiv2026

Classification of quaternionic skew-Hermitian symmetric spaces

We provide a complete classification of quaternionic skew-Hermitian symmetric spaces, namely symmetric spaces that admit a torsion-free ${\rm SO}^{*}(2n){\rm Sp}(1)$-structure for arbitrary $n>1$. Moreover, we prove that any homogeneous quaternionic skew-Hermitian manifold is necessarily a symmetric space.

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arXiv2026

Submanifolds of almost quaternionic skew-Hermitian manifolds

We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds $(M^{4n}, Q, ω)$, including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces.

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arXiv2024

Curvature of quaternionic skew-Hermitian manifolds and bundle constructions

This articles is devoted to a description of the second-order differential geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is, of quaternionic skew-Hermitian manifolds $(M, Q, ω)$. We provide a curvature characterization of such integrable geometric structures, based on the holonomy theory of symplectic connections and we study qualitative properties of the induced Ricci tensor. Then we proceed with bundle constructions over such a manifold $(M, Q, ω)$. In particular, we prove the existence of almost hypercomplex skew-Hermitian structures on the Swann bundle over $M$ and investigate their integrability.

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arXiv2023

Decomposable $(5,6)$-solutions in eleven-dimensional supergravity

We present decomposable (5,6)-solutions $\widetilde{M}^{1,4} \times M^6$ in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogue of work done by I. Chrysikos and A. Galaev who made similar computations for decomposable (6,5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with negative Einstein constant (in the "mostly plus" convention) and Riemannian Kähler-Einstein manifolds with positive Einstein constant. This conclusion generalizes a result of C. N. Pope and P. van Nieuwenhuizen concerning the appearance of six-dimensional Kähler-Einstein manifolds in eleven-dimensional supergravity. In this setting we construct infinitely many non-symmetric decomposable (5, 6)-supergravity backgrounds by using the infinitely many Lorentzian Einstein-Sasakian structures with negative Einstein constant on the 5-sphere, known from the work of C. P. Boyer et al.

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arXiv2023

Differential geometry of ${\mathsf{SO}}^{\ast}(2n)$-type structures -- Integrability

We study almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as the geometric structures underlying $\mathsf{SO}^\ast(2n)$- and $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures, respectively. The corresponding intrinsic torsions were computed in the previous article in this series, and the algebraic types of the geometries were derived, together with the minimal adapted connections (with respect to certain normalizations conditions). Here we use these results to present the related first-order integrability conditions in terms of the algebraic types and other constructions. In particular, we use distinguished connections to provide a more geometric interpretation of the presented integrability conditions and highlight some features of certain classes. The second main contribution of this note is the illustration of several specific types of such geometries via a variety of examples. We use the bundle of Weyl structures and describe examples of $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures in terms of functorial constructions in the context of parabolic geometries.

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arXiv2022

Reductive homogeneous Lorentzian manifolds

We study homogeneous Lorentzian manifolds $M = G/L$ of a connected reductive Lie group $G$ modulo a connected reductive subgroup $L$, under the assumption that $M$ is (almost) $G$-effective and the isotropy representation is totally reducible. We show that the description of such manifolds reduces to the case of semisimple Lie groups $G$. Moreover, we prove that such a homogeneous space is reductive. We describe all totally reducible subgroups of the Lorentz group and divide them into three types. The subgroups of Type I are compact, while the subgroups of Type II and Type III are non-compact. The explicit description of the corresponding homogeneous Lorentzian spaces of Type II and III (under some mild assumption) is given. We also show that the description of Lorentz homogeneous manifolds $M = G/L$ of Type I, reduces to the description of subgroups $L$ such that $M=G/L$ is an admissible manifold, i.e., an effective homogeneous manifold that admits an invariant Lorentzian metric. Whenever the subgroup $L$ is a maximal subgroup with these properties, we call such a manifold minimal admissible. We classify all minimal admissible homogeneous manifolds $G/L$ of a compact semisimple Lie group $G$ and describe all invariant Lorentzian metrics on them.

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arXiv2022

Differential geometry of $\mathsf{SO}^\ast(2n)$-type structures

We study $4n$-dimensional smooth manifolds admitting a $\mathsf{SO}^*(2n)$- or a $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure, where $\mathsf{SO}^*(2n)$ is the quaternionic real form of $\mathsf{SO}(2n, \mathbb{C})$. We show that such $G$-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better known almost hypercomplex/quaternionic-Hermitian structures (hH/qH for short). We present several equivalent definitions of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon's $\mathsf{E}\mathsf{H}$-formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our $G$-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of symmetric spaces $K/L$ with $K$ semisimple admitting an invariant torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure. This paper is the first in a series aiming at the description of the differential geometry of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures.

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arXiv2021

Ancient solutions of the homogeneous Ricci flow on flag manifolds

For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on $M$, and by [BöLS17] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold $M=G/K$ with second Betti number $b_{2}(M)=1$, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose $α$-limit set consists of fixed points at infinity of ${\mathscr{M}}^G$. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.

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arXiv2020

Homogeneous 8-manifolds admitting invariant Spin(7)-structures

We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fernández classification. Finally, we analyse the associated Spin(7)-connection with torsion.

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arXiv2020

Homogeneous Einstein metrics on non-Kähler C-spaces

We study homogeneous Einstein metrics on indecomposable non-Kählerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family \[ M_{\ell, m, n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad \ell, m, n\in\mathbb{Z}_{+} \] and examine several of its geometric properties. We show that invariant metrics on $M_{\ell, m, n}$ are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers $\ell, m, n$, by applying mapping degree theory we provide the existence of at least one $\mathsf{SU}(\ell+m+n)$-invariant Einstein metric on $M_{\ell, m, n}$. For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed $\ell, m, n$, we provide the numerical form of all non-isometric invariant Einstein metrics.

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arXiv2019

Decomposable (6, 5)-solutions in eleven-dimensional supergravity

This paper presents a series of constructions providing eleven-dimensional bosonic supergravity backgrounds. In particular, we treat Lorentzian manifolds given in terms of twisted products of six-dimensional Lorentzian manifolds and five-dimensional Riemannian manifolds. By considering a representative flux 4-form adapted to this setting, we analyse the system of bosonic supergravity equations and describe the corresponding geometric constraints. The new supergravity backgrounds appear for special cases associated to the adapted flux 4-form. For example, we provide a relation of eleven-dimensional supergravity with Ricci-isotropic Walker manifolds, and illustrate several results in their terms and in terms of Ricci-flat Riemannian manifolds.

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arXiv2019

A note on the volume of $\nabla$-Einstein manifolds with skew-torsion

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first variation of the volume on a compact Einstein manifold.

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arXiv2018

Decomposable $(4,7)$ solutions in eleven-dimensional supergravity

Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3, 1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold $(M^{7}, g)$. We describe a class of decomposable eleven-dimensional supergravity backgrounds on the product manifold $({\mathcal{M}}^{10, 1}=\widetilde{M}^{3,1} \times M^7, g_{\mathcal{M}}=\widetilde{g}+g)$, endowed with a flux form given in terms of the volume form on $\widetilde{M}^{3, 1}$ and a closed $4$-form $F^{4}$ on $M^{7}$. We show that the Maxwell equation for such a flux form can be read in terms of the co-closed 3-form $ϕ=\star_{7}F^{4}$. Moreover, the supergravity equation reduces to the condition that $(\widetilde{M}^{3,1},\widetilde{g})$ is an Einstein manifold with negative Einstein constant and $(M^7, g, F)$ is a Riemannian manifold which satisfies the Einstein equation with a stress-energy tensor associated to the 3-form $ϕ$. Whenever this 3-form is generic, the Maxwell equation induces a weak ${\rm G}_2$-structure on $M^{7}$ and then we obtain decomposable supergravity backgrounds given by the product of a weak ${\rm G}_2$-manifold $(M^7, ϕ, g)$ with a Lorentzian Einstein manifold $(\widetilde{M}^{3,1},\widetilde{g})$. We classify homogeneous 7-manifolds $M^{7}=G/H$ of a compact Lie group $G$ and indicate the cosets which admit an invariant or non-invariant ${\rm G}_2$-structure, or even no ${\rm G}_2$-structure. Then we construct examples of compact homogeneous Riemannian 7-manifolds endowed with non-generic invariant 3-forms which satisfy the Maxwell equation, but the construction of decomposable homogeneous supergravity backgrounds of this type remains an open problem.

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arXiv2017

A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications

Consider a Riemannian spin manifold $(M^{n}, g)$ $(n\geq 3)$ endowed with a non-trivial 3-form $T\inΛ^{3}T^{*}M$, such that $\nabla^{c}T=0$, where $\nabla^{c}:=\nabla^{g}+\frac{1}{2}T$ is the metric connection with skew-torsion $T$. In this note we introduce a generalized $\frac{1}{2}$-Ricci type formula for the spinorial action of the Ricci endomorphism ${\rm Ric}^{s}(X)$, induced by the one-parameter family of metric connections $\nabla^{s}:=\nabla^{g}+2sT$. This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrödinger-Lichnerowicz formula related to the square of the Dirac operator $D^{s}$, induced by $\nabla^{s}$, under the condition $\nabla^{c}T=0$. In the same case, we provide integrability conditions for $\nabla^{s}$-parallel spinors, $\nabla^{c}$-parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kähler manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 5, 6 and 7, respectively.

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arXiv2016

Spin structures on compact homogeneous pseudo-Riemannian manifolds

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

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arXiv2016

Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces

This paper is devoted to a systematic study and classification of invariant affine or metric connections on certain classes of naturally reductive spaces. For any non-symmetric, effective, strongly isotropy irreducible homogeneous Riemannian manifold $(M=G/K, g)$, we compute the dimensions of the spaces of $G$-invariant affine and metric connections. For such manifolds we also describe the space of invariant metric connections with skew-torsion. For the compact Lie group ${\rm U}_{n}$ we classify all bi-invariant metric connections, by introducing a new family of bi-invariant connections whose torsion is of vectorial type. Next we present applications related with the notion of $\nabla$-Einstein manifolds with skew-torsion. In particular, we classify all such invariant structures on any non-symmetric strongly isotropy irreducible homogeneous space.

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