On conformally flat contact metric manifolds

In the present paper we classify the conformally flat contact metric manifolds of dimension satisfying . We prove that these manifolds are Sasakian of constant curvature 1.     

On vertical skew symmetric almost contact 3-structures

We consider a (2m + 3)-dimensional Riemannian manifold M(ξ _r, η^r, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξ _r, η^r, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product of two totally geodesic submanifolds, where is a 2m-dimensional Kaehlerian submanifold and is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained.     

Equiaffine structures on statistical manifolds and Bayesian statistics

Relations between equiaffine geometry and Bayesian statistics are studied. A prior distribution in Bayesian statistics is regarded as a volume form on a statistical manifold. Applying equiaffine geometry to Bayesian statistics, the relation between alpha-parallel priors and the Jeffreys prior is given. As geometric results, conditions for a statistical submanifold to have an equiaffine structure are also given.     

Hypersurfaces in statistical manifolds

The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given as a first step of the statistical submanifold theory. A complex version of the notion of statistical structures is also introduced.     

On Hermitian curvature flow on almost complex manifolds

In the present paper we generalize the Hermitian curvature flow introduced and studied in Streets and Tian (2011) [6] to the almost complex case.     

Examples of almost Einstein structures on products and in cohomogeneity one

Almost Einstein manifolds are conformally Einstein up to a scale singularity, in general. This notion comes from conformal tractor calculus. In the current paper we discuss almost Einstein structures on closed Riemannian product manifolds and on 4-manifolds of cohomogeneity one. Explicit solutions are found by solving ordinary differential equations. In particular, we construct three families of closed 4-manifolds with almost Einstein structure corresponding to the boundary data of certain unimodular Lie groups. Two of these families are Bach-flat, but neither (globally) conformally Einstein nor half conformally flat. On products with a 2-sphere we find an exotic family of almost Einstein structures with hypersurface singularity as well.     

Statistical manifolds are statistical models

In this note we prove that any smooth (C¹ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C¹ resp.) statistical manifolds as finite dimensional statistical models.     

On ø-quasiconformally symmetric Sasakian manifolds

We study locally and globally ø-quasiconformally symmetric Sasakian manifolds. We show that a globally ø-quasiconformally symmetric Sasakian manifold is globally ø-symmetric. Some observations for a 3-dimensional locally ø-symmetric Sasakian manifold are given. We also give an example of a 3-dimensional locally ø-quasiconformally symmetric Sasakian manifold.     

Natural almost contact structures and their 3D homogeneous models

We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left‐invariant examples on three‐dimensional Lie groups, and show that any simply connected homogeneous Riemannian three‐manifold admits a natural almost contact structure having g as a compatible metric. Moreover, we investigate left‐invariant CR structures corresponding to natural almost contact metric structures.     

Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

A nullity condition for complex contact metric manifolds

A nullity condition for real contact manifolds is defined by Blair, Koufogiorgos and Papantoniu. Lately, Boeckx classified such manifolds completely. In this paper, a nullity condition for complex contact manifolds is defined as follows: take a complex contact manifold whose vertical space is annihilated by the curvature. Then, apply an $\mathcal{H}$-homothetic deformation. In this way, we get a condition which is invariant under $\mathcal{H}$-homothetic deformations. A complex contact manifold satisfying this condition is called a complex (,)-space. Some curvature properties of complex (,)-spaces are studied and it is shown that, just as in the real case, the curvature tensor of a complex (,)-space is completely determined.     

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains in ℝ^{2 n} . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.     

Left invariant contact structures on Lie groups

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.     

Some classes of almost contact metric manifolds and contact Riemannian submersions

Locally conformal almost quasi-Sasakian manifolds are related to the Chinea--Gonzales classification of almost contact metric manifolds. It follows that these manifolds set up a wide class of almost contact metric manifolds containing several interesting subclasses. Contact Riemannian submersions whose total space belongs to each of the considered classes are then investigated. The explicit expression of the integrability tensor and of the mean curvature vector field of each fibre are given. This allows us to state the integrability of the horizontal distribution and/or the minimality of the fibres in particular cases. The classes of the base space and of the fibres are also determined, so extending several well-known results.     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

Lorentzian Clairaut submersions

We investigate a class of semi-Riemannian submersions satisfying a Lorentzian analogue of the classical Clairaut's relation for surfaces of revolution. We show that a Lorentzian submersion with one-dimensional fibers is Clairaut if and only if the fibers are totally umbilic with a gradient field as the normal curvature vector field. We also investigate the behavior of timelike and null geodesics in Lorentzian Clairaut submersions. In particular, every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic in the base space with respect to a conformally related metric on the base space if and only if the integrability tensor of the submersion vanishes.     

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-K?hler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-K?hler is given.      

ParaHermitian and paraquaternionic manifolds

A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed.     

Tight contact structures with no symplectic fillings

We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings. Oblatum 7-XII-2000 & 14-XI-2001?Published online: 9 April 2002