arXiv subjects
Publications and source records attributed to Jorge Lauret.
We study both the complex and Hermitian geometry of C-spaces, i.e., compact homogeneous spaces M=G/K admitting a G-invariant complex structure. Some results on Hodge, Bott-Chern and Aeppli cohomologies are given. We also prove classification results in the pluriclosed, parallel Bismut torsion and locally conformally Kähler (or Vaisman) cases, respectively, as well as existence results for balanced and Calabi-Yau with torsion metrics.
Given a compact semisimple Lie group G and a maximal torus T of G, we give an explicit description of all left and Ad(T)-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on 2d+1 parameters in the irreducible case, where dim(T)=2d. As applications, we obtain that the only left and Ad(T)-invariant pluriclosed metrics which are also CYT are bi-invariant metrics (i.e., Bismut flat) and study the pluriclosed flow as a neat ODE system.
Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.
Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so called {\it aligned}, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of $G$-invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces.
Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/ΔK$, where $ΔK$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant Einstein metrics on $M$, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on $M$ for most spaces $H/K$ such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of $\mathfrak{k}$ is a multiple of the Killing form of $\mathfrak{h}$ (e.g., $K$ simple), a class which contains $17$ families and $50$ individual examples. A complete classification is obtained in the case when $H/K$ is an irreducible symmetric space with $K$ simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on $M=H\times H/ΔK$ (none of which is Einstein), obtaining that the standard metric is a global minimum.
A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and $ric(g)=\tfrac{1}{4} H_g^2$. On any homogeneous space $M=G/K$, where $G=G_1\times G_2$ is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat $G$-invariant generalized metric, which is proved to be unique among a $4$-parameter space of metrics in many cases, including when $K$ is neither abelian nor semisimple. On the other hand, if $K$ is simple and the standard metric is Einstein on both $G_1/π_1(K)$ and $G_2/π_2(K)$, we give a one-parameter family of Bismut Ricci flat $G$-invariant generalized metrics on $G/K$ and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form $M=G\times G/ΔK$ and for $M^{35}=SO(8)\times SO(7)/G_2$. A Corrigendum has been added in Appendix A.
The classification of compact homogeneous spaces of the form $M=G/K$, where $G$ is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are $4$ infinite families and $3$ isolated spaces found by Nikonorov and Rodionov in the 90s. In this paper, we prove that most of these standard Einstein metrics are unstable as critical points of the scalar curvature functional on the manifold of all unit volume $G$-invariant metrics on $M$, providing a lower bound for the coindex in the case of Ledger-Obata spaces. On the other hand, examples of stable (in particular, local maxima) invariant Einstein metrics on certain homogeneous spaces of non-simple Lie groups are also given.
The third real de Rham cohomology of compact homogeneous spaces is studied. Given $M=G/K$ with $G$ compact semisimple, we first show that each bi-invariant symmetric bilinear form $Q$ on $\mathfrak{g}$ such that $Q|_{\mathfrak{k}\times\mathfrak{k}}=0$ naturally defines a $G$-invariant closed $3$-form $H_Q$ on $M$, which plays the role of the so called Cartan $3$-form $Q([\cdot,\cdot],\cdot)$ on the compact Lie group $G$. Indeed, every class in $H^3(G/K)$ has a unique representative $H_Q$. Secondly, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., $b_3(G/K)=s-1$ if $G$ has $s$ simple factors, we give the conditions to be fulfilled by $Q$ and a given $G$-invariant metric $g$ in order for $H_Q$ to be $g$-harmonic, in terms of algebraic invariants of $G/K$. As an application, we obtain that any $3$-form $H_Q$ is harmonic with respect to the standard metric, although for any other normal metric, there is only one $H_Q$ up to scaling which is harmonic. Furthermore, among a suitable $(2s-1)$-parameter family of $G$-invariant metrics, we prove that the same behavior occurs if $\mathfrak{k}$ is abelian: either every $H_Q$ is $g$-harmonic (this family of metrics depends on $s$ parameters) or there is a unique $g$-harmonic $3$-form $H_Q$ (up to scaling). In the case when $\mathfrak{k}$ is not abelian, the special metrics for which every $H_Q$ is $g$-harmonic depend on $3$ parameters.
Back in 1985, Wang and Ziller obtained a complete classification of all homogeneous spaces of compact simple Lie groups on which the standard or Killing metric is Einstein. The list consists, beyond isotropy irreducible spaces, of 12 infinite families (two of them are actually conceptual constructions) and 22 isolated examples. We study in this paper the nature of each of these Einstein metrics as a critical point of the scalar curvature functional.
For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an application, we compute the $G$-invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with $b_2(M)=1$. This allows to deduce the $G$-stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.
Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature function. The case when g is naturally reductive is studied in special detail.
In this short note, three infinite families of neutrally stable homogeneous Einstein metrics are ruled out as candidates for local maxima of the Hilbert action.
The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M=G/K is studied. We focus on the metrics at which the Ricci curvature map is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.
We provide the second known example of an extremally Ricci pinched closed G2-structure on a compact 7-manifold, by finding a lattice in the only unimodular solvable Lie group admitting a left-invariant G2-structure. Furthermore, the Laplacian coflow and its solitons are studied on a 6-parameter family of left-invariant coclosed G2-structures on this Lie group. In this way, we obtain a 4-parameter subfamily of expanding solitons. The family is locally pairwise non-equivalent.
A complete classification of left-invariant closed G2-structures on Lie groups which are extremally Ricci pinched, up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the G2-structure is exact in all cases except one, given by the only example in which the Lie group is unimodular.
Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we obtain three new examples of extremally Ricci pinched G2-structures and that they are all necessarily steady Laplacian solitons. The deformation and rigidity of such structures are also studied.
The concept of soliton, in its most general version, allows us to find canonical or distinguished elements on any set provided with an equivalence relation and an `optimal' tangent direction at each point. We study in this paper solitons on homogeneous spaces, which have consolidated its role as a quite useful tool to find soliton geometric structures in Riemannian, pseudo-Riemannian, complex, symplectic and G2 geometries.
We give an overview of what is known on Lie groups admitting a left-invariant metric of negative Ricci curvature, including many natural questions and conjectures in the solvable case. We also introduce an open and convex cone C(n) of derivations attached to each nilpotent Lie algebra n, which is defined as the image of certain moment map and parametrizes a set of solvable Lie algebras with nilradical n admitting Ricci negative metrics.