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Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle / Luiz Carlos Barbosa Da Silva

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Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle / Luiz Carlos Barbosa Da Silva Universidade Federal de Pernambuco Centro de Ciências Exatas e da Natureza Departamento de Matemática Programa de Doutorado em Matemática Luiz Carlos Barbosa da Silva Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum arXiv:1806.08830v1 [math.DG] 22 Jun 2018 Mechanics of a Constrained Particle Recife 2017 Luiz Carlos Barbosa da Silva Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle Trabalho apresentado ao Programa de Pós- graduação em Matemática do Departamento de Matemática da Universidade Federal de Pernambuco como requisito parcial para a obtenção do grau de Doutor em Matemática. Supervisor: Prof. Dr. Fernando Antônio Nóbrega Santos Recife 2017 Catalogação na fonte Bibliotecária Monick Raquel Silvestre da S. Portes, CRB4-1217 D111d Da Silva, Luiz Carlos Barbosa Differential geometry of rotation minimizing frames, spherical curves, and quantum mechanics of a constrained particle / Luiz Carlos Barbosa da Silva. – 2017. 129 f.: il., fig. Orientador: Fernando Antônio Nóbrega Santos. Tese (Doutorado) – Universidade Federal de Pernambuco. CCEN, Matemática, Recife, 2017. Inclui referências. 1. Matemática. 2. Curva esférica. I. Santos, Fernando Antônio Nóbrega (orientador). II. Título. 510 CDD (23. ed.) UFPE- MEI 2017-210 LUIZ CARLOS BARBOSA DA SILVA DIFFERENTIAL GEOMETRY OF ROTATION MINIMIZING FRAMES, SPHERICAL CURVES, AND QUANTUM MECHANICS OF A CONSTRAINED PARTICLE Tese apresentada ao Programa de Pós- graduação do Departamento de Matemática da Universidade Federal de Pernambuco, como requisito parcial para a obtenção do título de Doutorado em Matemática. Aprovado em: 27/07/2017 BANCA EXAMINADORA ________________________________________________________ Prof. Dr. Fernando Antônio Nóbrega Santos (Orientador) Universidade Federal de Pernambuco _________________________________________________________ Prof. Dr. Eduardo Shirlippe Goes Leandro (Examinador Interno) Universidade Federal de Pernambuco ________________________________________________________ Prof. Dr. Antonio Fernando Pereira de Sousa (Examinador Interno) Universidade Federal de Pernambuco __________________________________________________________ Prof. Dr. Cristiano Costa Bastos (Examinador Externo) Universidade Federal Rural de Pernambuco __________________________________________________________ Prof.Dr. Bruno Carneiro da Cunha (Examinador Externo) Universidade Federal de Pernambuco To my mother Noemia for all the love and support. Acknowledgements It was March of 2008, the beginning of my journey as an undergrad student at the Universidade Federal de Pernambuco (UFPE). My dreams at that point? Well, I just wanted to learn as much as possible and then work as a scientist... I’m kidding, of course I also dreamed about winning a Nobel prize, mom would be really proud of me... but I promptly realized that it was a better idea to invest my time and efforts on solving the homework assignments! What a wonderful period... I remember that the beginning of each new semester at those initials years was accompanied by the same recurrent questions: am I smart enough to learn all these subjects? will I be happy doing this? is this my dream after all???... maybe I should throw it all away and go to Bollywood ... With time questions changed a little: does that professor hate me? will I have a job???... Well, ten years have passed since then and I do not ask myself such questions anymore, or at least not all of them. I am a person who finds pleasure in doing what I do and yes, I had a dream and I made it real... I am a scientist! Figure 1 – From PHD Comics <http://phdcomics.com/>. Now it is the time for new goals, new dreams, and new disappointments also. Maybe, I should try working and living in that not so beautiful city where one of my heroes worked too... or maybe teach that infamous course the way I think is the best one. Anyway, the point here is that I didn’t walk alone. On the road I met friends, good professors, not so good professors... advisors, collaborators, crazy and also reasonable referees, other researches... Well, science is also a human endeavor after all. In short, this is the moment to say thanks! I know it is a cliché to thank all the professors that one encounters along the way, but I will do it anyway. I would like to thank all the individuals that helped me along these long, wonderful, illuminating, and sometimes very difficult 10 years at UFPE. It was a great pleasure to meet and learn from you all. In particular, I would like to mention professors Antônio de Sousa, César Castilho, Eduardo Leandro, Francisco Brito, Hamid Hassanzadeh, Henrique Araújo, Maurício Coutinho-Filho, Ramón Mendoza, Sérgio Santa Cruz, Solange Rutz, and naturally my advisor Fernando Santos. I also thank the staff from the Department of Mathematics. I would like to take the opportunity to thank all the good discussions with professor Carmine Ortix from Utrecht University, the Netherlands, and professor Fernando Etayo from Universidad de Cantabria, Spain. I would also like to thank all my collaborators: Bertrand Berche and Sebastien Fumeron from Université de Lorraine, France; Fábio G. Ribeiro from Instituto Federal da Paraíba, Brazil; Cristiano C. Bastos, Fernando Moraes, and José Deibsom da Silva from Universidade Federal Rural de Pernambuco, Brazil; and finally my advisor Fernando A. N. Santos and former advisor Maurício D. Coutinho- Filho from UFPE. They all contributed a lot for my formation. In particular, professor Coutinho-Filho and his pursuit of a research done with all the rigor and care that it deserves influenced me a lot. I do my best to be as good and as professional as he is. Besides, I also want to express my gratitude to my advisor, professor Santos. We work together since I was an undergrad and then I must say that he decisively helped me in achieving my goals. In addition, his enthusiasm, support, friendship, and efforts to teach as much as possible about how it is like to be a scientist were decisive to me. I don’t have words to say how much I am grateful and I am not able to imagine a better advisor. An attentive reader will probably note that our Bibliography contains texts written in 5 different languages. This reflects the importance of language skills in the daily life of a scientist. So, I would like to take this opportunity and thank all the teachers that I had and have: in particular, Mariano Hebenbrock (German), Alexandra Kasky (German), Laura Trutt (French), and Alisa Zingerman (Hebrew). In this respect ,one should not forget the important role played by some institutions. So, I would like to thank the Núcleo de Línguas e Culturas from UFPE, the Centro Cultural Brasil Alemanha, and also Italki, HelloTalk, Memrise, and Duolingo. Probably, the ultimate cliché is that of acknowledging the family. But sometimes life is not that easy and the support from our relatives can not be taken for granted. So, I must say that I am a very luck guy for the opportunity of having a very supportive and special mother. She didn’t have the opportunity to proceed further with her education, but did everything she could in order to offer her children a different fate. Then, thank you mom for everything, your support was crucial to give me the necessary strength to go on in pursuit of my dreams. Last but not least, I would like to thank the rest of my family. In particular, my sister Luciana, my nephew Kléber, my grandparents Josefa, José and Severina. I also thank my girlfriend Bruna, the last two years were like a dream. In addition, some people say that friends are the family that we can choose. I would like to thank all my friends. Unfortunately, I don’t have a good memory to remember and mention here all their names. So, let me just thank some friends from the high school Leonardo, Leandro, and Thony; the friends who graduated with me in 2011 (BSc. in Math.) Edgar, Gilson, and João; Gabriel for being my tourist guide in Rome and for the useful discussion about geometry and mathematics in general; David and Frank for all the beers in Nancy and useful discussions about physics; and Renato, Alan, Jaime, Sally, and Rúbia for all the useful discussion about mathematics. Finally, let me thank the staff from Overleaf where I wrote this thesis and all my other manuscripts, Researchgate which keeps me updated about what my peers are doing, arXiv for its essential preprint service, and also Google Scholar for the excellent search engine and its track of citations. They definitely help in making science more democratic (in this respect, I would also like to thank other initiatives that fight for making research available for all). Lastly, let me also thank the financial support from the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the master and doctoral scholarships and also the Instituto Nacional de Matemática Pura e Aplicada (IMPA) for the financial support to participate in the XIX Brazilian School on Differential Geometry (2016) and XXXI Brazilian Mathematical Colloquium (2017). “La filosofia naturale è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi, io dico l’universo, ma non si può intendere se prima non s’impara a intender la lingua e conoscer i caratteri nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto” - Il Saggiatore, Galileo Galilei (1564-1642) Abstract This thesis is devoted to the differential geometry of curves and surfaces along with applications in quantum mechanics. In the 1st part we initially introduce the well known Frenet frame and then discuss on plane curves with a power-law curvature. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any moving frame, from which one defines Rotation Minimizing (RM) frames as those frames that achieve this minimum.
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