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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 September 24, 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. 24 Sep 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 Sep 24]. 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

University of Oxford

2.
Rubio, Roberto.
Generalized *geometry* of type Bn.

Degree: PhD, 2014, University of Oxford

URL: http://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669803

Generalized geometry of type B_{n} is the study of geometric structures in T+T<sup>*</sup>+1, the sum of the tangent and cotangent bundles of a manifold and a trivial rank 1 bundle. The symmetries of this theory include, apart from B-fields, the novel A-fields. The relation between B_{n}-geometry and usual generalized geometry is stated via generalized reduction. We show that it is possible to twist T+T<sup>*</sup>+1 by choosing a closed 2-form F and a 3-form H such that dH+F^{2}=0. This motivates the definition of an odd exact Courant algebroid. When twisting, the differential on forms gets twisted by d+Fτ+H. We compute the cohomology of this differential, give some examples, and state its relation with T-duality when F is integral. We define B_{n}-generalized complex structures (B_{n}-gcs), which exist both in even and odd dimensional manifolds. We show that complex, symplectic, cosymplectic and normal almost contact structures are examples of B_{n}-gcs. A B_{n}-gcs is equivalent to a decomposition (T+T<sup>*</sup>+1)<sub>ℂ</sub>= L + L + U. We show that there is a differential operator on the exterior bundle of L+U, which turns L+U into a Lie algebroid by considering the derived bracket. We state and prove the Maurer-Cartan equation for a B_{n}-gcs. We then work on surfaces. By the irreducibility of the spinor representations for signature (n+1,n), there is no distinction between even and odd B_{n}-gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+U-cohomology. We finish by defining G^{2}_{2}-structures on 3-manifolds, a structure with no analogue in usual generalized geometry. We prove an analogue of the Moser argument and describe the cone of G^{2}_{2}-structures in cohomology.

Subjects/Keywords: 516; Mathematics; 3-manifold; almost contact geometry; complex geometry; deformation theory; G2(2)-structure; generalized complex geometry; twisted cohomology; generalized geometry; Lie algebroid

Record Details Similar Records

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

APA (6^{th} Edition):

Rubio, R. (2014). Generalized geometry of type Bn. (Doctoral Dissertation). University of Oxford. Retrieved from http://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669803

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

Rubio, Roberto. “Generalized geometry of type Bn.” 2014. Doctoral Dissertation, University of Oxford. Accessed September 24, 2020. http://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669803.

MLA Handbook (7^{th} Edition):

Rubio, Roberto. “Generalized geometry of type Bn.” 2014. Web. 24 Sep 2020.

Vancouver:

Rubio R. Generalized geometry of type Bn. [Internet] [Doctoral dissertation]. University of Oxford; 2014. [cited 2020 Sep 24]. Available from: http://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669803.

Council of Science Editors:

Rubio R. Generalized geometry of type Bn. [Doctoral Dissertation]. University of Oxford; 2014. Available from: http://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669803

Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ)

3. Θεοφανίδης, Θεοχάρης. Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.

Degree: 2011, Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ)

URL: http://hdl.handle.net/10442/hedi/27049

J. de Dios Perez, F. G. Santos and Y. J. Suh in [29], studied real hypersurfaces
of dimension greater than 3 in complex projective spaces, whose
Jacobi structure operator is of Codazzi type. In chapter 2 we study real
hypersurfaces under the same condition, fulfilling the case of hyperbolic
spaces of dimension n > 3 as long as the case of 3dimensional
hypersurfaces.
M. Ortega, J. de Dios Perez and F. G. Santos in [24] studied real hypersurfaces
of dimension greater than 3, in complex space forms, whose
Jacobi structure operator is parallel. J. de Dios Perez and F. G.Santos in
[27] studied real hypersurfaces of dimension greater than 3 with recurrent
structure Jacobi operator. In chapter 3 we improve [27] in dimension 3, by
studying real hypersurfaces with D recurrent
structure Jacobi operator,
in complex planes. Furthermore we improve [24] by studying real hypersurfaces
of dimension n > 3 with recurrent structure Jacobi operator.
J. T. Cho and U H.
Ki in [13] classified real hypersurfaces of dimension
greater than 3, in complex projective spaces, which satisfy the conditions
l = l and lA = Al everywhere in the real hypersurface M. In chapter 4
we improve the previous paper by classifying real hypersurfaces in complex
space forms of dimension 2n (n 2) satisfying the condition l = l in D
and the condition lA = Al either in D or in D?. Moreover we classify real
hypersurfaces in complex space forms of dimension 2n (n 2) satisfying
the condition l = l in D and the condition (r l) = , 2 C1 either in
D or in D?.

Subjects/Keywords: Διαφορική γεωμετρία; Πολλαπλότητα Riemann; Μιγαδικός χώρος μορφής; Πραγματική υπερεπιφάνεια; Δομή σχεδόν επαφής; Τελεστής δομής Jacobi; Differential geometry; Riemannian manifolds; Complex space form; Real hypersurface; Almost contact structure; Jacobi structure operator

Record Details Similar Records

❌

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

APA (6^{th} Edition):

Θεοφανίδης, . . (2011). Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. (Thesis). Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ). Retrieved from http://hdl.handle.net/10442/hedi/27049

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):

Θεοφανίδης, Θεοχάρης. “Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.” 2011. Thesis, Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ). Accessed September 24, 2020. http://hdl.handle.net/10442/hedi/27049.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Θεοφανίδης, Θεοχάρης. “Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.” 2011. Web. 24 Sep 2020.

Vancouver:

Θεοφανίδης . Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. [Internet] [Thesis]. Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ); 2011. [cited 2020 Sep 24]. Available from: http://hdl.handle.net/10442/hedi/27049.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Θεοφανίδης . Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. [Thesis]. Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ); 2011. Available from: http://hdl.handle.net/10442/hedi/27049

Not specified: Masters Thesis or Doctoral Dissertation