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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

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

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 December 05, 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. 05 Dec 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 Dec 05]. 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

2.
Pro, Curtis.
On *Riemannian* *Submersions* and Diffeomorphism Stability.

Degree: Mathematics, 2012, University of California – Riverside

URL: http://www.escholarship.org/uc/item/2z16d2kf

This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside. Abstract of chapter II: If π:M → B is a Riemannian Submersion and M has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.Abstract of Chapter III: Though Riemannian submersions preserve non-negative sectional curvature this does not generalize to Riemannian submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian submersion π: M → B for which {Ricci}_{p}(M)>0 and at some point p∈ B, {Ricci}_{p}(B)<0. Abstract of Chapter IV: The smallest r so that a metric r – ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature ≥ k and radius ≤ r. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

Subjects/Keywords: Mathematics; Diffeomorphsim Stability; Dual Foliations; Isometric Group Actions; Ricci Curvature; Riemannian Submersions

…*Riemannian* *Submersions* Need Not Preserve Positive Ricci Curvature . . .
1.3 The Diffeomorphism Type… …1
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3 *Riemannian* *Submersions* Need Not Preserve Positive… …that *Riemannian* *submersions* preserve nonnegative curvature. In addition, if either term on… …nonnegatively curved metrics so that π and Q : P × F → P ×G F = E become *Riemannian*
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One might ask if something…

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Pro, C. (2012). On Riemannian Submersions and Diffeomorphism Stability. (Thesis). University of California – Riverside. Retrieved from http://www.escholarship.org/uc/item/2z16d2kf

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

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

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Thesis, University of California – Riverside. Accessed December 05, 2020. http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Web. 05 Dec 2020.

Vancouver:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Internet] [Thesis]. University of California – Riverside; 2012. [cited 2020 Dec 05]. Available from: http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Thesis]. University of California – Riverside; 2012. Available from: http://www.escholarship.org/uc/item/2z16d2kf

Not specified: Masters Thesis or Doctoral Dissertation