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1. Polymerakis, Kleanthis. Rigidity and deformability of immersed submanifolds.

Degree: 2019, University of Ioannina; Πανεπιστήμιο Ιωαννίνων

URL: http://hdl.handle.net/10442/hedi/46312

We study the Bonnet problem for surfaces in 4-dimensional space forms Q4c . Two isometric surfaces are said to have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. Noncongruent surfaces with the same mean curvature are called Bonnet mates. A surface in Q4c is called a Bonnet, or a proper Bonnet surface, if it admits either at least one, or infinitely many Bonnet mates, respectively. We introduce the notions of isotropically isothermic and strongly isotropically isothermic surfaces in Q4c as a generalization of the notion of isothermic surfaces in Q3c and we show that isotropic isothermicity is a conformally invariant property. We show that if a non-compact simply connected surface f : M → Q4c is not proper Bonnet, then it admits either at most one Bonnet mate, or exactly three. If such a surface is proper Bonnet, then the moduli space M(f) of congruence classes of all isometric immersions of M into Q4c that have the same mean curvature with f, is diffeomorphic to a manifold. Proper Bonnet surfaces are distinguished in two categories: the tight surfaces whose moduli space is 1-dimensional with at most two connected components diffeomorphic to S1 ≃ R/2πZ, and the flexible ones whose moduli space is diffeomorphic to the torus S1 × S1. We prove that isotropic isothermicity characterizes proper Bonnet surfaces and in particular, strong isotropic isothermicity characterizes the flexible surfaces. Moreover, we show that a half totally non isotropically isothermic surface is always a Bonnet surface which in particular, admits exactly three Bonnet mates if it is furthermore strongly totally non isotropically isothermic. We also prove that a Bonnet surface lying in a totally geodesic hypersurface of Q4c with non-constant mean curvature, admits at least two Bonnet mates that do not lie in any totally umbilical hypersurface of Q4c. We prove that if both Gauss lifts of a compact surface to the twistor bundle are not vertically harmonic, then the surface admits at most three Bonnet mates. In particular, we show that such a surface admits at most one Bonnet mate, under additional assumptions involving isotropic isothermicity. We show that non-minimal surfaces with a vertically harmonic Gauss lift possess a holomorphic quadratic differential, yielding thus a Hopf-type theorem. We prove that such surfaces allow locally a 1-parameter family of isometric deformations with the same mean curvature. This family is trivial only if the surface is superconformal. For such compact surfaces with non-parallel mean curvature, we prove that the moduli space is the disjoint union of two sets, each one being either finite, or a circle. In particular, for surfaces in R4 we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.

Μελετάμε το πρόβλημα Bonnet για επιφάνειες σε τετραδιάστατους χώρους μορφής Q4c. Δύο ισομετρικές επιφάνειες λέγεται ότι έχουν την ίδια μέση…

Subjects/Keywords: Μέση καμπυλότητα; Απεικόνιση Gauss; Ολόμορφο διαφορικό; Πρόβλημα Bonnet; Mean curvature; Gauss map; Holomorphic differential; Bonnet problem

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

APA (6^{th} Edition):

Polymerakis, K. (2019). Rigidity and deformability of immersed submanifolds. (Thesis). University of Ioannina; Πανεπιστήμιο Ιωαννίνων. Retrieved from http://hdl.handle.net/10442/hedi/46312

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):

Polymerakis, Kleanthis. “Rigidity and deformability of immersed submanifolds.” 2019. Thesis, University of Ioannina; Πανεπιστήμιο Ιωαννίνων. Accessed September 25, 2020. http://hdl.handle.net/10442/hedi/46312.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Polymerakis, Kleanthis. “Rigidity and deformability of immersed submanifolds.” 2019. Web. 25 Sep 2020.

Vancouver:

Polymerakis K. Rigidity and deformability of immersed submanifolds. [Internet] [Thesis]. University of Ioannina; Πανεπιστήμιο Ιωαννίνων; 2019. [cited 2020 Sep 25]. Available from: http://hdl.handle.net/10442/hedi/46312.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Polymerakis K. Rigidity and deformability of immersed submanifolds. [Thesis]. University of Ioannina; Πανεπιστήμιο Ιωαννίνων; 2019. Available from: http://hdl.handle.net/10442/hedi/46312

Not specified: Masters Thesis or Doctoral Dissertation

Texas Tech University

2.
Kose, Zeynep S.
GEOMETRIC AND NUMERICAL METHODS FOR *BONNET* PROBLEMS AND SURFACE CONSTRUCTION.

Degree: Mathematics and Statistics, 2010, Texas Tech University

URL: http://hdl.handle.net/2346/ETD-TTU-2010-12-1130

This PhD dissertation studies several important problems in differential geometry and its numerical applications.
Its novel contributions to these areas include the following:
1) solving Bonnet problems using the Cartan theory of moving frames, Cartan structure equations, and numerical analysis;
2) studying surface duality in Riemannian geometries, in terms of a certain duality between mean curvature and the Hopf differential factor;
3) determining sufficient geometric conditions for a surface to admit isothermic coordinates;
4) constructing surfaces that admit isothermic coordinates; determining isothermic coordinate charts when starting from an arbitrary chart; implementing these methods numerically.
The moving frames and Cartan structure equations are written in terms of the first and second fundamental forms, and the Lax system is consequently reinterpreted; orthonormal moving frames are obtained as solutions to the Bonnet-Lax system, via numerical integration. Certain classifications of families of surfaces are studied in terms of the first and second fundamental forms, with respect to certain prescribed invariants. Numerical methods are applied to this theoretical framework in order to solve Bonnet problems, construct isothermic coordinate charts for surfaces that admit them, and construct dual surfaces (Christoffel transforms). Several visual examples are provided, as well as the corresponding numerical methods and code.
*Advisors/Committee Members: Toda, Magdalena D. (Committee Chair), Aulisa, Eugenio (committee member), Ibragimov, Akif (committee member), Iyer, Ram V. (committee member).*

Subjects/Keywords: Bonnet problem; Dual surfaces; Curvature line coordinates; Surface construction; Surface visualization

Record Details Similar Records

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

APA (6^{th} Edition):

Kose, Z. S. (2010). GEOMETRIC AND NUMERICAL METHODS FOR BONNET PROBLEMS AND SURFACE CONSTRUCTION. (Thesis). Texas Tech University. Retrieved from http://hdl.handle.net/2346/ETD-TTU-2010-12-1130

Not specified: Masters Thesis or Doctoral Dissertation

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

Kose, Zeynep S. “GEOMETRIC AND NUMERICAL METHODS FOR BONNET PROBLEMS AND SURFACE CONSTRUCTION.” 2010. Thesis, Texas Tech University. Accessed September 25, 2020. http://hdl.handle.net/2346/ETD-TTU-2010-12-1130.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Kose, Zeynep S. “GEOMETRIC AND NUMERICAL METHODS FOR BONNET PROBLEMS AND SURFACE CONSTRUCTION.” 2010. Web. 25 Sep 2020.

Vancouver:

Kose ZS. GEOMETRIC AND NUMERICAL METHODS FOR BONNET PROBLEMS AND SURFACE CONSTRUCTION. [Internet] [Thesis]. Texas Tech University; 2010. [cited 2020 Sep 25]. Available from: http://hdl.handle.net/2346/ETD-TTU-2010-12-1130.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kose ZS. GEOMETRIC AND NUMERICAL METHODS FOR BONNET PROBLEMS AND SURFACE CONSTRUCTION. [Thesis]. Texas Tech University; 2010. Available from: http://hdl.handle.net/2346/ETD-TTU-2010-12-1130

Not specified: Masters Thesis or Doctoral Dissertation