Almost quaternionic structures on compact rank one symmetric spaces
We prove that the only CROSSes that admit a (not necessarily homogeneous) almost quaternionic structure are ${\mathbb{H}} {\mathbb{P}}^n$ and ${\mathbb{C}} {\mathbb{P}}^2$.
math.AT↗
arXiv subjects
Eugenia Loiudice
Publications and source records attributed to Eugenia Loiudice.
GKM actions on almost quaternionic manifolds
We introduce quaternionic structures on abstract GKM graphs, as the combinatorial counterpart of almost quaternionic structures left invariant by a torus action of GKM type. In the GKM$_3$ setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM$_3$ actions on positive quaternion-Kähler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract GKM$_3$ graph with quaternionic structure satisfying this restriction on the 2-faces of the GKM graph is that of a torus action on quaternionic projective space ${\mathbb{H}} P^n$ or the Grassmannian ${\mathrm{Gr}}_2({\mathbb{C}}^n)$ of complex 2-planes in ${\mathbb{C}}^n$.
math.DG↗
Para-Sasakian $ϕ$-symmetric spaces
We study the Boothby-Wang fibration of para-Sasakian manifolds and introduce the class of para-Sasakian $ϕ$-symmetric spaces, canonically fibering over para-Hermitian symmetric spaces. Using this fibration we give a method to explicitly construct semisimple para-Sasakian $ϕ$-symmetric spaces. We provide moreover an example of non-semisimple para-Sasakian $ϕ$-symmetric space.
math.DG↗
GKM actions on cohomogeneity one manifolds
We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group diagram of $M$ such that the $T$-action on $M$ is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples.
math.DG↗
How to construct all metric $f$-$K$-contact manifolds
We show that any compact metric $f$-$K$-contact, respectively $S$-manifold is obtained from a compact $K$-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.
math.DG↗
Metric $f$-contact manifolds satisfying the $(κ,μ)$-nullity condition
We prove that if the $f$-sectional curvature at any point $p$ of a $(2n+s)$-dimensional $f$-$(κ,μ)$ manifold with $n>1$ is independent of the $f$-section at $p$, then it is constant on the manifold. Moreover, we also prove that an $f$-$(κ,μ)$ manifold which is not an $S$-manifold is of constant $f$-sectional curvature if and only if $μ=κ+1$ and we give an explicit expression for the curvature tensor field. Finally, we present some examples.
math.DG↗
On the topology of metric $f$-$K$-contact manifolds
We observe that the class of metric $f$-$K$-contact manifolds, which naturally contains that of $K$-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric $f$-$K$-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.
math.DG↗
Canonical fibrations of contact metric $(κ,μ)$-spaces
We present a classification of the complete, simply connected, contact metric $(κ,μ)$-spaces as homogeneous contact metric manifolds, by studying the base space of their canonical fibration. According to the value of the Boeckx invariant, it turns out that the base is a complexification or a para-complexification of a sphere or of a hyperbolic space. In particular, we obtain a new homogeneous representation of the contact metric $(κ,μ)$-spaces with Boeckx invariant less than $-1$.
math.DG↗
A dimensional restriction for a class of contact manifolds
In this work we consider a class of contact manifolds $(M,η)$ with an associated almost contact metric structure $(ϕ, ξ, η,g)$. This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class $C_9\oplus C_{10}$ defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension $4n+1$. Under the assumption that the sectional curvature of the horizontal $2$-planes is constant at one point, we obtain that these manifolds must have dimension $5$.
math.DG↗