The differential geometry of the fibres of an almost contract metric submersion

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space.Mathematical SciencesD. Phil. (Mathematics

Clairaut Submersion

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion

Cohomological aspects of non-Kähler manifolds

In this thesis, we study cohomological properties of non-Kähler manifolds. In particular, we are concerned in investigating the cohomology of compact (almost-)complex manifolds, and of manifolds endowed with special structures, e.g., symplectic structures, $\mathbf{D}$-complex structures in the sense of F. R. Harvey and H. B. Lawson, exhaustion functions satisfying positivity conditions. In Chapter 0, which contains no original material, we collect the basic notions concerning almost-complex, complex, and symplectic structures, we recall the main results on Hodge theory for Kähler manifolds, and we summarize the classical results on deformations of complex structures, on currents and de Rham homology, and on solvmanifolds. In Chapter 1, we study cohomological properties of compact complex manifolds, and in particular the Bott-Chern cohomology. By using exact sequences introduced by J. Varouchas, we prove a Frölicher-type inequality for the Bott-Chern cohomology, which also provides a characterization of the validity of the $\partial\overline{\partial}$-Lemma in terms of the dimensions of the Bott-Chern cohomology groups. We then prove a Nomizu-type result for the Bott-Chern cohomology, showing that, for certain classes of complex structures on nilmanifolds, the Bott-Chern cohomology is completely determined by the associated Lie algebra endowed with the induced linear complex structure. As an application, we explicitly study the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations. Finally, we study the Bott-Chern cohomology of complex orbifolds of the type X/G, where X is a compact complex manifold and G a finite group of biholomorphisms of X. In Chapter 2, we study cohomological properties of almost-complex manifolds. Firstly, we recall the notion of $\mathcal{C}^\infty$-pure-and-full almost-complex structure, which has been introduced by T.-J. Li and W. Zhang in order to investigate the relations between the compatible and the tamed symplectic cones on a compact almost-complex manifold and with the aim to throw light on a question by S. K. Donaldson. In particular, we are interested in studying when certain subgroups, related to the almost-complex structure, let a splitting of the de Rham cohomology of an almost-complex manifold, and their relations with cones of metric structures. Then, we focus on $\mathcal{C}^\infty$-pure-and-fullness on several classes of (almost-)complex manifolds, e.g., solvmanifolds endowed with left-invariant almost-complex structures, semi-Kähler manifolds, almost-Kähler manifolds. Then, we study the behaviour of $\mathcal{C}^\infty$-pure-and-fullness under small deformations of the complex structure and along curves of almost-complex structures, investigating properties of stability, and of semi-continuity for the dimensions of the invariant and anti-invariant subgroups of the de Rham cohomology with respect to the almost-complex structure. Then, we consider the cone of semi-Kähler structures on a compact almost-complex manifold and, in particular, by adapting the results by D. P. Sullivan on cone structures, we compare the cones of balanced metrics and of strongly-Gauduchon metrics on a compact complex manifold. In Chapter 3, we study the cohomological properties of (differentiable) manifolds endowed with special structures, other than (almost-)complex structures. More precisely, we investigate the cohomology of symplectic manifolds; then, we study cohomological decompositions on $\mathbf{D}$-complex manifolds in the sense of F. R. Harvey and H. B. Lawson; finally, we consider domains in $\mathbb{R}^n$ endowed with a smooth proper strictly p-convex exhaustion function, and, using $\mathrm{L}^2$ -techniques, we give another proof of a consequence of J.-P. Sha’s theorem, and of H. Wu’s theorem, on the vanishing of the higher degree de Rham cohomology groups

© Hindawi Publishing Corp. SKEW-SYMMETRIC VECTOR FIELDS ON A CR-SUBMANIFOLD OF A PARA-KÄHLERIAN MANIFOLD

We deal with a CR-submanifold M of a para-Kählerian manifold ˜M, which carries a J-skewsymmetric vector field X. It is shown that X defines a global Hamiltonian of the symplectic form Ω on M ⊤ and JX is a relative infinitesimal automorphism of Ω. Other geometric properties are given. 2000 Mathematics Subject Classification: 53C15, 53C20, 53C21. 1. Introduction. CR-submanifolds