Differential Geometry and Its Visualization

Differential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences. Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors’ own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them. Features Extensive, full colour visualisations Numerous exercises Self-contained and comprehensive treatment of the topic

Table of Contents:
1. Curves in Three–dimensional Euclidean Space. 1.1. Points and Vectors. 1.2. Vector–valued Functions of a Real Variable. 1.3. The General Concept of Curves. 1.4. Some Examples of Planar Curves. 1.5. The Arc Length of a Curve. 1.6. The Vectors of the Trihedron of a Curve. 1.7. Frenet’s Formulae. 1.8. The Geometric Significance of Curvature and Torsion. 1.9. Osculating Circles and Spheres. 1.10. Involutes and Evolutes. 1.11. The Fundamental Theorem of Curves. 1.12. Lines of Constant Slope. 1.13. Spherical Images of a Curve. 2. Surfaces in Three–dimensional Euclidean Space. 2.1. Surfaces and Curves on Surfaces. 2.2. The Tangent Planes and Normal Vectors of a Surface. 2.3. The Arc Length, Angles and Gauss’s First Fundamental Coefficients. 2.4. the Curvature of Curves on Surfaces, Geodesic and Normal Curvature. 2.5. The Normal, Principal, Gaussian and Mean Curvature. 2.6. The Shape of a Surface in the Neighbourhood of a Point. 2.7. Dupin’s Indicatrix. 2.8. Lines of Curvature and Asymptotic Lines. 2.9. Triple Orthogonal Systems. 2.10. the Weingarten Equations. 3. The Intrinsic Geometry of Surfaces. 3.1. the Christoffel Symbols. 3.2. Geodesic Lines. 3.3. Geodesic Lines on Surfaces with Orthogonal Parameters. 3.4. Geodesic Lines on Surfaces of Revolution. 3.5. the Minimum Property of Geodesic Lines. 3.6. Orthogonal and Geodesic Parameters. 3.7. Levi–civitá Parallelism. 3.8. Theorema Egregium. 3.9. Maps Between Surfaces. 3.10. the Gauss–bonnet Theorem. 3.11. Minimal Surfaces. 4. Tensor Algebra and Riemannian Geometry. 4.1. Differentiable Manifolds. 4.2. Transformation of Bases. 4.3. Linear Functionals and Dual Spaces. 4.4. Tensors of Second Order. 4.5. Symmetric Bilinear Forms and Inner Products. 4.6. Tensors of Arbitary Order. 4.7. Symmetric and Anti–symmetric Tensors. 4.8. Riemann Spaces. 4.9. the Christoffel Symbols. 5. Tensor Analysis. 5.1. Covariant Differentiation. 5.2. the Covariant Derivative of an (R, S)–tensor. 5.3. the Interchange of Order for Covariant Differentiation and Ricci’s Identity. 5.4. Bianchi’s Identities for the Covariant Derivative of the Tensors of Curvature. 5.5. Beltrami’s Differentiators. 5.6. a Geometric Meaning of the Covariant Differentiation, the Levi–civitá Parallelism. 5.7. The Fundamental Theorem for Surfaces. 5.8. A Geometric Meaning of the Riemann Tensor of Curvature. 5.9. Spaces With Vanishing Tensor of Curvature. 5.10. An Extension of Frenet’s Formulae. 5.11. Riemann Normal Coordinates and the Curvature of Spaces.

About the Author :
Eberhard Malkowsky is a Full Professor of Mathematics in retirement at the State University of Novi Pazar in Serbia. He earned his Ph.D. degree and habilitation at the Department of Mathematics of the Justus-Liebig Universität Giessen in Germany in 1982 and 1988, respectively. He was a professor of mathematics at universities in Germany, South Africa, Jordan, Turkey and Serbia, and a visiting professor in the USA, India, Hungary and France. Furthermore, he participated as an invited or keynote speaker with more than 100 lectures in international scientific conferences and congresses. He is a member of the editorial boards of twelve journals of international repute. His list of publications contains 175 research papers in international journals. He is the author or co-author of nine books, and the editor or co-editor of six proceedings of international conferences. He supervised 6 Ph.D. theses and a great number of B.Sc. and M.Sc. theses in mathematics. His research and work areas include functional analysis, differential geometry and software development for the visualization of mathematics. Ćemal Dolićanin is a Professor Emeritus at the Department of Sciences and Mathematics at the State University of Novi Pazar in Serbia and a member of the Serbian Academy of Non-linear Sciences. He was dean of the Electro-Technical Faculty and the Faculty of Technical Sciences in Priština, vice-rector of the University of Priština and founder and rector of the State University of Novi Pazar, Serbia. He obtained his M.Sc. degree at the Faculty of Mathematics of the University of Belgrade in 1974, and his Ph.D. degree at the Faculty of Sciences and Mathematics of the University of Priština in 1980. He published more than 20 books, 23 papers in national scientific journals, 41 papers in international scientific journals, and gave more than 50 lectures at international scientific conferences. He was a visiting professor in Germany, Belorussia and Russia. He supervised 14 PhD theses and a great number of B.Sc. and M.Sc. theses in mathematics. His research and work areas include Euclidean and non-Euclidean geometry, differential geometry and applied mathematics. He is very active in promoting mathematics, and has established the Center for the Advancement and Popularization of Mathematics at the State University of Novi Pazar. He participated in the implementation of several national scientific projects TEMPUS projects and was the coordinator of Master's study programs with the World University Service Austria (WUS). Vesna Veličković is a Professor at the Department of Computer Science at the Faculty of Sciences and Mathematics of the University of Niš, Serbia. She obtained her magister degree at the Faculty of Mathematics of the University of Belgrade in 1996, and her Ph.D. degree at the Faculty of Sciences and Mathematics of the University of Niš in 2012. She published 3 books, 30 papers in international scientific journals, 12 papers in national scientific journals, and gave more than 50 lectures at international scientific conferences. She participated in 3 international scientific and 4 software projects, and 7 study visits in Serbia, Germany, Bulgaria, Romania and Turkey. Together with Professor Malkowsky, she is developing the software MV-Graphics for visualization of mathematics. With large-format graphics, they participated in three exhibitions of mathematical art. For ten years she worked with pupils talented in mathematics and programming. Her students won a number of medals at International Olympiads in Informatics. She is still very active in organizing contests in programming and promoting of mathematics. Her research areas are software development, computer graphics and visualization of mathematics.