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University of South Africa

1.
Tshikunguila, Tshikuna-Matamba.
The differential geometry of the *fibres* of an almost contract metric submersion
.

Degree: 2013, University of South Africa

URL: http://hdl.handle.net/10500/18622

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.
*Advisors/Committee Members: Batubenge, T. A (advisor), Massamba, F (advisor).*

Subjects/Keywords: Differential Geometry; Riemannian submersions; Almost contact metric submersions; CR-submersions; Contact CR-submanifolds; Almost contact metric manifolds; Almost Hermitian manifolds; Riemannian curvature tensor; Holomorphic sectional curvature; Minimal fibres; Superminimal fibres; Umbilicity

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APA (6^{th} Edition):

Tshikunguila, T. (2013). The differential geometry of the fibres of an almost contract metric submersion . (Doctoral Dissertation). University of South Africa. Retrieved from http://hdl.handle.net/10500/18622

Chicago Manual of Style (16^{th} Edition):

Tshikunguila, Tshikuna-Matamba. “The differential geometry of the fibres of an almost contract metric submersion .” 2013. Doctoral Dissertation, University of South Africa. Accessed October 28, 2020. http://hdl.handle.net/10500/18622.

MLA Handbook (7^{th} Edition):

Tshikunguila, Tshikuna-Matamba. “The differential geometry of the fibres of an almost contract metric submersion .” 2013. Web. 28 Oct 2020.

Vancouver:

Tshikunguila T. The differential geometry of the fibres of an almost contract metric submersion . [Internet] [Doctoral dissertation]. University of South Africa; 2013. [cited 2020 Oct 28]. Available from: http://hdl.handle.net/10500/18622.

Council of Science Editors:

Tshikunguila T. The differential geometry of the fibres of an almost contract metric submersion . [Doctoral Dissertation]. University of South Africa; 2013. Available from: http://hdl.handle.net/10500/18622