MATH 557 — Differential Geometry

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MATH457/557 Differential Geometry Course Outline Topics # of weeks Chapter 1: Curves in the plane and in the space 0.5 Chapter 2: Curvature of a planar curve, curvature and torsion of a space curve, geometric geometric interpretation of torsion, Frenet-Serret equations and their applications 1.5 Chapter 3: The Isoperimetric Inequality, the Four Vertex Theorem 0.5 Chapter 4: Surfaces in 3D: important examples, smooth surfaces and maps, tangent vector and space, derivative of a smooth map between surfaces, normals and orientability 1.0 Chapter 5: Classification of quadratic surfaces, ruled surface, surface of revolution, triply orthogonal systems, applications of the Inverse Function Theorem 1.5 Chapter 6: The first fundamental form of a surface, length and area on a surface, isometries of surfaces, conformal and equiareal mappings of surfaces, the Archimedes' Theorem 1.5 Chapters 7 and 8: Curvature of surfaces: the second fundamental form, the Gauss and Weingarten maps, the normal and geodesic curvatures, the Meusnier's Theorem, parallel transport and covariant derivative, Gauss equations, Gaussian, mean, and principal curvature, surfaces of constant Gaussian and mean curvature, flat surfaces 2.0 Chapter 9: Definition of geodesics and its basic properties, geodesic equations, geodesics on surfaces of revolution, geodesic coordinates 1.5 Chapter 10: The Codazzi-Mainardi equations, the Gauss' Theorem, surfaces of constant Gaussian curvature, geodesic mappings 1.5 Chapter 13: The Gauss-Bonnet Theorem (for simple closed curves, curvilinear polygons, and polygons, and compact surfaces) and its applications, the Map Colouring Theorem, holonomy and Gaussian curvature, singularities of vector fields, critical points 1.5 Exams: 1.0 Textbook: Elementary Differential Geometry, 2nd edition by Andrew Pressley Approved ___________ .

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Differential Geometry: Curvature and Holonomy Austin Christian
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics.
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