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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-K\"ahler 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 ${\mathbb{C}}^n$., Comment: 35 pages

We study the Boothby-Wang fibration of para-Sasakian manifolds and introduce the class of para-Sasakian $\phi$-symmetric spaces, canonically fibering over para-Hermitian symmetric spaces. Using this fibration we give a method to explicitly construct semisimple para-Sasakian $\phi$-symmetric spaces. We provide moreover an example of non-semisimple para-Sasakian $\phi$-symmetric space., Comment: 30 pages

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., Comment: 18 pages, 6 figures, 3 tables. Minor changes, in particular additional explanations were added in the proof of Proposition 2.3. Proposition 3.6 added. Results are unchanged

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.

We prove that if the $f$-sectional curvature at any point $p$ of a $(2n+s)$-dimensional $f$-$(\kappa,\mu)$ 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 $f$-$(\kappa,\mu)$ manifold which is not an $S$-manifold is of constant $f$-sectional curvature if and only if $\mu=\kappa+1$ and we give an explicit expression for the curvature tensor field. Finally, we present some examples.

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., Comment: 16 pages. v2: added Remark 4.6 and a few references

We present a classification of the complete, simply connected, contact metric $(\kappa,\mu)$-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 $(\kappa,\mu)$-spaces with Boeckx invariant less than $-1$.

In this work we consider a class of contact manifolds $(M,\eta)$ with an associated almost contact metric structure $(\phi, \xi, \eta,g)$. 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 $5$., Comment: 9 pages, accepted for publication in Demonstratio Mathematica