Mathematics

It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu (κ,μ) -manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic (κ,μ) -spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic (κ,μ) -spaces. There is obtained local classification of paracontact metric (κ,μ) -spaces, and almost paracosymplectic (κ,μ) -spaces, for every possible value of κ .

A comparison among different constructions of the quaternionic 4 -form Φ Sp(2)Sp(1) and of the Cayley calibration Φ Spin(7) shows that one can start for them from the same collections of "Kähler 2-forms", entering in dimension 8 both in quaternion Kähler and in Spin(7) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16 , similar constructions allow to write explicit formulas for the canonical 4 -forms Φ Spin(8) and Φ Spin(7)U(1) , associated with Clifford systems related with the subgroups Spin(8) and Spin(7)U(1) of SO(16) . We characterize the calibrated 4 -planes of the 4 -forms Φ Spin(8) and Φ Spin(7)U(1) , extending in two different ways the notion of Cayley 4 -plane to dimension 16 .

A study is made of left-invariant G 2 -structures with an exact 3-form on a Lie group G whose Lie algebra g admits a codimension-one nilpotent ideal h . It is shown that such a Lie group G cannot admit a left-invariant closed G 2 -eigenform for the Laplacian and that any compact solvmanifold ??�G arising from G does not admit an (invariant) exact G 2 -structure. We also classify the seven-dimensional Lie algebras g with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact G 2 -structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras h admitting an exact SL(3,C) -structure ? or a half-flat SU(3) -structure (?,?) with exact ? , respectively.

In this paper we consider closed SL(3,C) -structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to G 2 -manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed SL(3,C) -structure and those which admit an invariant mean convex half-flat SU(3) -structure. We also prove that, if a solvmanifold admits an invariant tamed closed SL(3,C) -structure, then it also has an invariant symplectic half-flat SU(3) -structure.

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

A compact complex manifold V is called Vaisman if it admits an Hermitian metric which is conformal to a Kähler one, and a non-isometric conformal action by C . It is called quasi-regular if the C -action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of V . It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as a quasi-regular quotient of V . We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as an S 1 -quotient of M .

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

As a sequel to \cite{Licollapsing}, we study Calabi-Yau metrics collapsing along a holomorphic fibration over a Riemann surface. Assuming at worst canonical singular fibres, we prove a uniform diameter bound for all fibres in the suitable rescaling. This has consequences on the geometry around the singular fibres.

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.