Elementary topics in differential geometry pp 190209 cite as. Balazs csik os differential geometry e otv os lor and university faculty of science typotex 2014. Connections, curvature, and characteristic classes. Is there any particular books papers regarding the application of the theorem in the theory of general relativity. The gaussbonnet theorem is a profound theorem of differential geometry, linking global and local geometry.
An introduction to gaussian geometry sigmundur gudmundsson lund university. The idea is illustrated here in the example when p is. While the main topics are the classics of differential geometry the definition and geometric meaning of gaussian curvature, the theorema egregium, geodesics, and the gaussbonnet theorem the treatment is modern and studentfriendly, taking direct routes to explain, prove and apply the main results. Further topics holonomy and the gaussbonnet theorem, hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean. The gaussbonnet theorem, or gaussbonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry in the sense of curvature to their topology in the sense of the euler characteristic. We prove a discrete gaussbonnetchern theorem which states where summing the curvature over all vertices of a finite graph gv,e gives the euler characteristic of g. One of the deepest theorems in the differential geometry of surfaces is the gaussbonnet theorem. Free differential geometry books download ebooks online. Differential geometry a first course in curves and. A first course in curves and surfaces free book at e books directory. It rst discusses the language necessary for the proof and applications of a powerful generalization of the fundamental theorem of calculus, known as stokes theorem in rn. Some good books on application of gaussbonnetchern theorem. Differential geometry of curves and surfaces springerlink.
Do carmo, differential geometry of curves and surfaces. An introduction to differential forms, stokes theorem and gaussbonnet theorem anubhav nanavaty abstract. Then the gaussbonnet theorem, the major topic of this book, is discussed at great length. Introduction the generalized gauss bonnet theorem of allendoerferweil 1 and chern 2 has played an important role in the development of the relationship between modern differential geometry and algebraic topology, providing in particular. Modern differential geometry of curves and surfaces with mathematica, 2nd. Here are some differential geometry books which you might like to read while youre waiting for my dg book to be written. I doubt you will have entire books devoted to the subject, as you hoped for. This paper serves as a brief introduction to di erential geometry. The only prerequisites are one year of undergraduate calculus and linear algebra. Curves and surfaces graduate studies in mathematics. Riemann curvature tensor and gausss formulas revisited in index free notation. For readers bound for graduate school in math or physics, this is a clear.
Along the way we encounter some of the high points in the history of differential geometry, for example, gauss theorema egregium and the gaussbonnet theorem. A first course in differential geometry by woodward. On the other hand, kleiner and lott posted a paper on the arxiv in which they use ricci flow to geometrise 3orbifolds, so orbifolds certainly seem like a good arena in which to generalise differential geometry. I understand a 3rd volume is being written that will be more of a research monograph treating advanced topics and recent research developments. But its deepest consequence is the link between geometry and topology established by the gaussbonnet theorem. This is the 2dimensional version of the gaussbonnet theorem.
Elementary differential geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. That is, some books dont define abstract manifolds. Modern differential geometry of curves and surfaces with mathematica, 2nd ed. Aug 07, 2015 here we study the proof of the gauss bonnet theorem based on a rectangularization of a compact oriented surface. Differential geometry a first course in curves and surfaces. Exercises throughout the book test the reader s understanding of the material and sometimes illustrate extensions of the theory. I am currently doing an undergraduate project about gauss bonnet chern theorem. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. Requiring only multivariable calculus and linear algebra, it develops students geometric intuition through interactive computer graphics applets supported by sound theory. Let q be a compact orbifold of dimension n with boundary m, and let e.
Differential geometry of curves and surfaces, second edition takes both an analyticaltheoretical approach and a visualintuitive approach to the local and global properties of curves and surfaces. Aspects of differential geometry i synthesis lectures on. The theorem is a most beautiful and deep result in differential geometry. Is there any particular book suggestions regarding the application of the theorem in the theory of general relativity. The gauss bonnet theorem, or gauss bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. The wide range of topics includes curve theory, a detailed study of surfaces, curvature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the gaussbonnet theorem. Math 501 differential geometry herman gluck thursday march 29, 2012 7.
These ideas and many techniques from differential geometry have applications in physics, chemistry, materials. Along the way, the author discusses the exponential map, parallel transport, jacobi fields, minimal surfaces, spherical and hyperbolic geometry, cartography, gauss divergence theorem, and the gaussbonnet theorem. Already one can see the connection between local and global geometry. The gaussbonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry. Is there any good reference on the application of gauss bonnet chern theorem for fourdimensional manifold on. It is named after carl friedrich gauss, who was aware of a version of the theorem but never published it, and pierre ossian bonnet, who published a special case in 1848. One of the deepest theorems in the differential geometry of surfaces is the gauss bonnet theorem. Consider a surface patch r, bounded by a set of m curves. We prove a discrete gauss bonnet chern theorem which states where summing the curvature over all vertices of a finite graph gv,e gives the euler characteristic of g. The gaussian curvature of a surface is its dominant geometric property and it enters into almost every geometrical investigation. Calculus of variations and surfaces of constant mean curvature 107 appendix. The curvature of a compact surface completely determines its topological structure. The authors did not shy away from sophisticated topics.
The fundamental concept underlying the geometry of curves is the arclength of a parametrized curve. The textbook is a concise and well organized treatment of. Local theory, holonomy and the gauss bonnet theorem, hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature. These were used as the basic text on geometry throughout the western world for about 2000 years. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. Prerequisites are kept to an absolute minimum nothing beyond first courses in linear algebra and multivariable calculus and the most direct and straightforward approach is used. It yields a relation between the integral of the gaussian curvature over a given oriented closed surface s and the topology of s in terms of its euler number. Cherngaussbonnet is a bit of a benchmark for higherdimensional riemannian geometry, and id like to know the state of the art is in.
Nov 14, 2019 chapter 4 starts with a simple and elegant proof of stokes theorem for a domain. Differential geometry in physics by gabriel lugo university of north carolina at wilmington these notes were developed as a supplement to a course on differential geometry at the advanced undergraduate level, which the author has taught. An introductory textbook on the differential geometry of curves and surfaces in threedimensional euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. Please note that the content of this book primarily consists of articles. Are differential equations and differential geometry related. Gauss s formulas, christoffel symbols, gauss and codazzimainardi equations, riemann curvature tensor, and a second proof of gauss s theorema egregium. Latin text and various other information, can be found in dombrowskis book 1. Differential geometry of curves and surfaces, second.
Its most important version relates the average of the gaussian curvature to a property of the surface called its euler number which is topological, i. Covariant differentiation, parallel translation, and geodesics surfaces. The ten chapters of hicks book contain most of the mathematics that has become the standard background for not only differential geometry, but also much of modern theoretical physics and cosmology. The last chapter addresses the global geometry of curves, including periodic space curves and the fourvertices theorem for plane curves that are not necessarily convex. The gauss number characterizes geometry of p while the euler number characterizes the combinatorics of p. Elementary differential geometry andrew pressley download.
Bonnet in a special form for surfaces homeomorphic to a disc. A first course in differential geometry by woodward, lyndon. The course will conclude with various forms of the gaussbonnet theorem. Copies are available from the maths office, the electronic version can be found on duo. Local theory, holonomy and the gaussbonnet theorem, hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature. Based on serretfrenet formulae, the theory of space curves is developed and concluded with a detailed discussion on fundamental existence theorem. I am currently doing an undergraduate project about gaussbonnetchern theorem. Hicks van nostrand a concise introduction to differential geometry. Gauss bonnet theorem for surfaces and selected introductory topics in special and general relativity are also analyzed. The theorem says that for every polyhedron p, the gauss number of p the euler number of p. Course description this course is an introduction to differential geometry of curves and surfaces in three dimensional euclidean space. Introduction to differential geometry 1 from wolfram.
A first course is an introduction to the classical theory of space curves and surfaces offered at the graduate and post graduate courses in mathematics. The simplest case of gb is that the sum of the angles in a planar triangle is 180 degrees. Here are however two papers in which some version of the gaussbonnet theorem unfortunately not the 4 dimensional version makes interesting appearances. The gaussbonnet theorem has also been generalized to riemannian polyhedra. Gausss major published work on differential geometry is contained in the dis quisitiones. Chapter 1 introduction around 300 bc euclid wrote the thirteen books of the elements. This book is a comprehensive introduction to differential forms. Gaussbonnet theorem an overview sciencedirect topics. Riemann curvature tensor and gauss s formulas revisited in index free notation. Differential geometry course notes ebooks directory. Then the gauss bonnet theorem, the major topic of this book, is discussed at great length.
See robert greenes notes here, or the wikipedia page on gaussbonnet, or perhaps john lees riemannian manifolds book. Chern gauss bonnet is a bit of a benchmark for higherdimensional riemannian geometry, and id like to know the state of the art is in. Thus combinatorics of a polyhedron puts constraints on geometry of this polyhedron, and conversely, geometry. The gauss bonnet theorem the gauss bonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. The gaussbonnet theorem the gaussbonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. Nov 29, 2018 while the main topics are the classics of differential geometry the definition and geometric meaning of gaussian curvature, the theorema egregium, geodesics, and the gauss bonnet theorem the treatment is modern and studentfriendly, taking direct routes to explain, prove and apply the main results. Although we will not follow any of these strictly, the material can be found in them.
Introduction the generalized gaussbonnet theorem of allendoerferweil 1 and chern 2 has played an important role in the development of the relationship between modern differential geometry and algebraic topology, providing in particular. The gaussbonnet theorem, or gaussbonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their. The gaussbonnet theorem or gaussbonnet formula in differential. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the riemannian metric 4, 6, 7. Gausss formulas, christoffel symbols, gauss and codazzimainardi equations, riemann curvature tensor, and a second proof of gausss theorema egregium. Several results from topology are stated without proof, but we establish almost all. These are my rough, offthecuff personal opinions on the usefulness of some of the dg books on the market at this time. The proofs will follow those given in the book elements of differential. Part of the undergraduate texts in mathematics book series utm. A nice feature of this book are the pictures of the mathematicians who proved the relevant theorems makes mathematics have a human face. Hints for most of the 124 exercises appear at the back of the book. Exercises throughout the book test the readers understanding of the material. Many details my major professor assumed i knew are given here. Along the way we encounter some of the high points in the history of differential geometry, for example, gauss theorema egregium and the gauss bonnet theorem.
The course will conclude with various forms of the gauss bonnet theorem. Gaussbonnet and other major theorems are proven from first principles. Cohnvossen, some problems of differential geometry in the large, moscow 1959 in russian 4. Differential geometry of curves and surfaces springer. Exercises throughout the book test the readers understanding of the material and sometimes illustrate extensions of the theory. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and selfstudy. The topic mixes chromatic graph theory, integral geometry and is motivated by results known in differential geometry like the farymilnor theorem of 1950 which writes total curvature of a knot as an index expectation and is elementary. This texts has an early introduction to differential forms and their applications to physics. A discrete version of the gaussbonnet theorem um math. We used this book for our first semester in differential geometry. The gaussbonnet theorem is obviously not at the beginning of the. The gaussbonnet theorem is the most beautiful and profound result in the theory of surfaces.
A first course in curves and surfaces preliminary version summer, 2016. This is a textbook on differential geometry wellsuited to a variety of courses on this topic. Differential equations and differential geometry certainly are related. Gaussbonnet theorem for surfaces and selected introductory topics in special and general relativity are also analyzed. The topic mixes chromatic graph theory, integral geometry and is motivated by results known in differential geometry like the farymilnor theorem of 1950 which writes total curvature of a knot as an.
31 928 831 523 1340 869 513 1363 630 1411 1215 1481 1019 912 734 1261 1363 1393 1083 1396 1346 149 879 315 147 186 1198 499 572 303 1361 893 1221