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 Sciences / D. Phil. (Mathematics)

Les objects d'étude principaux de la thèse "Flots géométriques kähleriens et presque-kähleriens" sont des généralisations du flot de Calabi et du flot hermitienne de Yang--Mills. <p> Le flot de Calabi $partial_t omega = -i delbar del S(omega) =- i delbar del Lambda_omega <p> ho(omega) $ tente de déformer une forme initiale kählerienne vers une forme kählerienne $omega_c$ de courbure scalaire constante caractérisée par $S(omega_c) = Lambda_{omega_c} <p> ho(omega_c) = underline{S}$ dans la même classe de cohomologie. La généralisation étudiée est le flot de Calabi twisté qui remplace la forme de Kähler--Ricci $ho$ par $ho + alpha(t)$, où le emph{twist} $alpha(t)$ est une famille de $2$-formes qui converge vers $alpha_infty$. Le but de ce flot est de trouver des métriques kähleriennes $omega_{tc}$ de courbure scalaire twistées constantes caractérisées par $Lambda_{omega_{tc}} (ho(omega_{tc}) +alpha_infty) = underline{S} + underline{alpha}_infty$. L'existence et la convergence de ce flot sont établies sur des surfaces de Riemann à condition que le twist soit défini négatif et reste dans une classe de cohomologie fixe. <p>Si $E$ est un fibré véctoriel holomorphe sur une varieté kählerienne $(X,omega)$, une métrique de Hermite--Einstein $h_{he}$ est caractérisée par la condition $Lambda_omega i F_{he} = lambda id_E$. Le flot hermitien de Yang--Mills donné par $h^{-1}partial_t h =- [Lambda_omega iF_{h} - lambda id_E]$ tente de déformer une métrique hermitienne initiale vers une métrique Hermite--Einstein. La version classique du flot fixe la forme kählerienne $omega$. Le cas où $omega$ varie dans sa classe de cohomologie et converge vers $omega_infty$ est considéré dans la thèse. Il est démontré que le flot existe pour tout $t$ sur des surfaces de Riemann et converge vers une métrique Hermite--Einstein (par rapport à $omega_infty$) si le fibré $E$ est stable. <p> Les généralisations du flot de Calabi et du flot hermitien de Yang--Mills ne sont pas arbitraires, mais apparaissent naturellement comme une approximation du flot de Calabi sur des fibrés adiabatiques. Si $Z,X$ sont des variétés complexes compactes, $pi colon Z \ / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)