arXiv subjects
Publications and source records attributed to Fabio Podestà.
We prove that a compact Kähler manifold carrying a regular Riemannian foliation whose leaves are Lagrangian and minimal is flat. In order to prove this result, we establish a foliated version of an integral formula due to Ros.
We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on $\mathbb{R}^3$ that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.
In this article, we investigate the class of Hermitian manifolds whose Bismut connection has parallel torsion ({\rm BTP} for brevity). In particular, we focus on the case where the manifold is (locally) homogeneous with respect to a group of holomorphic isometries and we fully characterize the compact Chern flat {\rm BTP} manifolds. Moreover we show that certain compact flag manifolds are {\rm BTP} if and only if the metric is Kähler or induced by the Cartan-Killing form and we then characterize {\rm BTP} invariant metrics on compact semisimple Lie groups which are Hermitian w.r.t. a Samelson structure and are projectable along the Tits fibration. We state a conjecture concerning the question when the Bismut connection of a BTP compact Hermitian locally homogeneous manifold has parallel curvature, giving examples and providing evidence in some special cases.
Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order $4$ and (up to coverings) can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.
We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky-Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.
We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.
Given any non-compact real simple Lie group G of inner type and even dimension, we prove the existence of an invariant complex structure J and a Hermitian balanced metric with vanishing Chern scalar curvature on G and on any compact quotient $M=G/Γ$, with $Γ$ a cocompact lattice. We also prove that (M,J) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.
We prove the existence of a one-parameter family of nearly parallel $G_2$-structures on the manifold $S^3\times \mathbb R^4$, which are mutually non isomorphic and invariant under the cohomogeneity one action of the group $SU(2)^3$. This family connects the two locally homogeneous nearly parallel $G_2$-structures which are induced by the homogeneous ones on the sphere $S^7$.
We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite exstinction time $T>0$ and we analyze its behaviour when $t\to T$. We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.
Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost Kähler structure $(ω,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $ρ= λω$ for some $λ\in\mathbb R$, where $ρ$ is the Ricci form associated to the Chern connection.
We prove that the automorphism group of a compact 6-manifold $M$ endowed with a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension bounded by min$\{5,b_1(M)\}$. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on $T\mathbb{S}^3$ which are invariant under a cohomogeneity one action of SO(4).
We study the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel G$_2$-structure, showing that its identity component is abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G$_2$-structure. We also discuss some relevant examples.
We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result allowing to show the non-existence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This last condition is characterized in the general case.
We consider non-Kaehler compact complex manifolds which are homogeneous under the action of a compact Lie group of biholomorphisms and we investigate the existence of special (invariant) Hermitian metrics on these spaces. We focus on a particular class of such manifolds comprising the case of Calabi-Eckmann manifolds and we prove the existence of an invariant Hermitian metric which is Chern-Einstein, namely whose second Ricci tensor of the associated Chern connection is a positive multiple of the metric itself. The uniqueness is also discussed.
In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,ξ)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a compact simple Lie group, and $ξ\in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^*$ and the flag manifold structure of Ad(G)$ξ$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo--riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $ξ$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.
Given a simply connected compact generalized flag manifold M together with its invariant Kähler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the space of invariant Kähler metrics. We give sufficient and necessary conditions so that the metric g is a critical point for this functional. Moreover we prove that when M is a full flag manifold, the metric g is critical if and only if M= SU(3)/T^2 and in this case g is a maximum.
For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the Z_2-homology of a sphere.
We prove that any invariant hypercomplex structure on a homogeneous space $M = G/L$ where $G$ is a compact Lie group is obtained via the Joyce's construction, provided that there exists a hyper-Hermitian naturally reductive invariant metric on $M$.