arXiv subjects
Publications and source records attributed to Alejandro Tolcachier.
We study the smallest positive eigenvalue $\lambda_1$ of the Laplace-Beltrami operator associated with any compact strongly isotropy irreducible space. We provide an explicit expression for all simply connected cases. Furthermore, every strongly isotropy irreducible space is automatically an Einstein manifold, and we prove for each of them that $E<\lambda_1\leq 16E$, where $E$ denotes the corresponding Einstein constant.
Paul Schwahn recently exhibited 112 non-symmetric, connected, simply connected, compact Einstein manifolds that are stable with respect to the total scalar curvature functional restricted to the space of Riemannian metrics with constant scalar curvature and fixed volume. This stability follows from the inequality $\lambda_L > 2E$, where $\lambda_L$ denotes the smallest eigenvalue of the Lichnerowicz Laplacian on TT-tensors and $E$ is the corresponding Einstein factor. In this paper, we estimate the smallest positive eigenvalue $\lambda_1$ of the Laplace-Beltrami operator for connected, simply connected, non-symmetric standard Einstein manifolds $(G/H,g_{\operatorname{st}})$ with $G$ a compact and connected simple Lie group. We obtain that $\lambda_1>2E$ for all of them excepting $7$ spaces. As a consequence of our estimates, we establish that all stable Einstein manifolds found by Schwahn are in fact linearly stable with respect to Perelman's $\nu$-entropy.
It is known that there exist complex solvmanifolds $(\Gamma\backslash G,J)$ whose canonical bundle is trivialized by a holomorphic section which is not invariant under the action of $G$. The main goal of this article is to classify the six-dimensional Lie algebras corresponding to such complex solvmanifolds, thus extending the previous work of Fino, Otal and Ugarte for the invariant case. To achieve this, we complete the classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures and identify among them, the ones admitting complex structures with Chern-Ricci flat metrics. Finally we construct complex solvmanifolds with non-invariant holomorphic sections of their canonical bundle. In particular, we present an example of one such solvmanifold that is not biholomorphic to a complex solvmanifold with an invariant section of its canonical bundle. Additionally, we discover a new $6$-dimensional solvable strongly unimodular Lie algebra equipped with a complex structure that has a non-zero holomorphic $(3,0)$-form.
The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\operatorname{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\operatorname{SL}(2n+1,\mathbb{R})$, for all $n\in \mathbb{N}$. We then show that there are no HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, providing thus a new example of an Obata holonomy group properly contained in $\operatorname{GL}(m,\mathbb{H})$ and not contained in $\operatorname{SL}(m,\mathbb{H})$, where $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$.
An almost abelian Lie group is a solvable Lie group with a codimension-one normal abelian subgroup. We characterize almost Hermitian structures on almost abelian Lie groups where the almost complex structure is harmonic with respect to the Hermitian metric. Also, we adapt the Gray-Hervella classification of almost Hermitian structures to the family of almost abelian Lie groups. We provide several examples of harmonic almost complex structures in different Gray-Hervella classes on some associated compact almost abelian solvmanifolds.
We study complex solvmanifolds $Γ\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form $ψ$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of $ψ$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_α)$ is trivial only for $α=1$, so that $M$ is not an $\operatorname{SL}(n,\mathbb{H})$-manifold.
It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures $(J_{a,b},g_{a,b})$. We show in this article that the complex structure $J_{a,b}$ is harmonic with respect to $g_{a,b}$, i.e. it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or $k$-Gauduchon ($k\geq 2$). Finally, we study the Bismut connection associated to $(J_{a,b}, g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor $\operatorname{Ric}^B$ and the Bismut-Ricci form $ρ^B$. We show that these tensors vanish if and only if each Sasakian factor is $η$-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.
A flat solvmanifold is a compact quotient $Γ\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $Γ$ is a lattice of $G$. Any such Lie group can be written as $G=\mathbb{R}^k\ltimes_ϕ \mathbb{R}^m$ with $\mathbb{R}^m$ the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting $G$ by a lattice $Γ$ that can be decomposed as $Γ=Γ_1\ltimes_ϕΓ_2$, where $Γ_1$ and $Γ_2$ are lattices of $\mathbb{R}^k$ and $\mathbb{R}^m$, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.
In this article we study the relation between flat solvmanifolds and $G_2$-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of $\mathsf{GL}(n,\mathbb{Z})$ for $n=5$ and $n=6$. Then, we look for closed, coclosed and divergence-free $G_2$-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free $G_2$-structure whose finite holonomy is cyclic and contained in $G_2$, and examples of compact flat manifolds admitting a divergence-free $G_2$-structure.
This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group $\mathbb{Z}_n$ coincides with the minimal dimension of a compact flat manifold with holonomy group $\mathbb{Z}_n$. Finally, we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; exhibiting in the latter case a general construction to show examples of non cyclic holonomy groups.