The gauss image of the common edge shared by the faces. The eulerpoincar e number is the earliest invariant of algebraic topology. Containing the compulsory course of geometry, its particular impact is on elementary topics. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. The main proof was presented here the paper is behind a paywall, but there is a share link from elsevier, for a few days. Historical development of the gaussbonnet theorem article pdf available in science in china series a mathematics 514. 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. Several results from topology are stated without proof, but we establish almost all. Equilateral triangle, perpendicular bisector, angle bisector, angle made by lines, the regular hexagon, addition and subtraction of lengths, addition and subtraction of angles, perpendicular lines, parallel lines and angles, constructing parallel lines, squares and other. It arises as the special case where the topological index is defined in terms of betti numbers and the analytical index is defined in terms of the gaussbonnet integrand as with the twodimensional gaussbonnet theorem, there are generalizations when m is a manifold with boundary. This is a localglobal theorem par excellence, because it asserts the equality of two very differently defined quantities on a compact, orientable riemannian 2manifold m. Lectures on gaussbonnet richard koch may 30, 2005 1 statement of the theorem in the plane according to euclid, the sum of the angles of a triangle in the euclidean plane is equivalently, the sum of the exterior angles of a triangle is 2. Bonnet theorem, which asserts that the total gaussian curvature of a compact oriented 2dimensional riemannian manifold is independent of the riemannian metric.
In this paper i introduce and examine properties of discrete surfaces in e ort to prove a discrete gaussbonnet analog. This is an informal survey of some of the most fertile ideas which grew out of the attempts to better understand the meaning of this remarkable theorem. Gaussbonnet for discrete surfaces sohini upadhyay abstract. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero equivalently by definition, the theorem states that the field of complex numbers is algebraically closed. The gaussbonnet theorem, like few others in geometry, is the source of many fundamental discoveries which are now part of the everyday language of the modern geometer.
The naturality of the euler class means that when changing the riemannian metric, one stays in the same cohomology class. Theorem gauss s theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. There is evidence that descartes knew about this formula a century before euler, s. Mountains, earthquakes, and the gaussbonnet theorem the gaussbonnet theorem says that for any surface, total curvature 2 if an earthquake creates a new mountain tomorrow, generating additional positive curvature at the top of the mountain, that new positive curvature has to be exactly balanced by new negative curvature elsewhere. Since it is a topdimensional differential form, it is closed. The left hand side is the integral of the gaussian curvature over the manifold. See robert greenes notes here, or the wikipedia page on gaussbonnet, or perhaps john lees riemannian manifolds book. The following expository piece presents a proof of this theorem, building. The idea of proof we present is essentially due to. It is intrinsically beautiful because it relates the curvature of a manifolda geometrical objectwith the its euler characteristica topological one. The gauss bonnet chern theorem on riemannian manifolds yin li abstract this expository paper contains a detailed introduction to some important works concerning the gauss bonnet chern theorem. Riemann curvature tensor and gauss s formulas revisited in index free notation. The gaussbonnet theorem is an important theorem in differential geometry. The goal of these notes is to give an intrinsic proof of the gau.
The right hand side is some constant times the euler characteristic. Millman and parker 1977 give a standard differentialgeometric proof of the gaussbonnet theorem, and. Let s be a closed orientable surface in r 3 with gaussian curvature k and euler characteristic. 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. Lectures on the geometry of manifolds download pdf. We are finally in a position to prove our first major localglobal theorem in riemannian geometry. State and prove the gaussbonnet theorem for a spherical polygon with geodesic sides. It is named after the two mathematicians carl friedrich gau. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the riemannian metric 4, 6, 7. Mathematics volume 51, pages 777 784 2008 cite this article. That is, some books dont define abstract manifolds. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book.
Of course identifying this alternating sum with the alternating sum of the betti numbers of m, the so called morse equality, of necessity does require homological arguments. The total gaussian curvature of a closed surface depends only on the topology of the surface and is equal to 2. Featured texts all books all texts latest this just in smithsonian libraries fedlink us genealogy lincoln collection. From foucaults pendulum to the gaussbonnet theorem. The gaussbonnet theorem says that, for a closed 7 manifold. Euclidean geometry by rich cochrane and andrew mcgettigan. Historical development of the gaussbonnet theorem hunghsi wu 1 science in china series a. These notions of curvature tell us roughly what a surface looks like both locally and globally. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. The fundamental theorem of algebra states that every nonconstant singlevariable polynomial with complex coefficients has at least one complex root. In this lecture we introduce the gaussbonnet theorem. The levicivita connection is presented, geodesics introduced, the jacobi operator is discussed, and the gaussbonnet theorem is proved. Jm jdm here p is the section of sm over dm given by the outward unit normal vector. The gaussbonnet theorem is a theorem that connects the geometry of a shape with its topology.
Integrals add up whats inside them, so this integral represents the total amount of curvature of the manifold. The gaussbonnetchern theorem on riemannian manifolds. Gausss formulas, christoffel symbols, gauss and codazzimainardi equations, riemann curvature tensor, and a second proof of gausss theorema egregium. The gaussbonnet theorem is obviously not at the beginning of the. To state the general gaussbonnet theorem, we must first define curvature. Bonnet s theorem on the diameter of an oval surface. The gaussbonnet theorem is a special case when m is a 2d manifold. Its importance lies in relating geometrical information of a surface to a purely topological characteristic, which has resulted in varied and powerful applications.
Search the history of over 424 billion web pages on the internet. We present a selfcontained proof of the gaussbonnet theorem for twodimensional surfaces embedded in. This proof can be found in guillemin and pollack 1974. Gaussbonnet is a deep result in di erential geometry that illustrates a fundamental relationship between the curvature of a surface and its euler characteristic. It is a vast generalization of a formula involving convex polyhedra due to euler. In this case, dffa djo3, where d3 is the exterior derivative for the submanifold nj. Also, the proofs are for surfaces embedded in 3space, although the theorems hold. The gaussbonnetchern theorem on riemannian manifolds arxiv.
No matter which choices of coordinates or frame elds are used to compute it, the gaussian curvature is the same function. The gaussbonnet theorem is one of the most beautiful and one of the deepest. The first part, analytic geometry, is easy to assimilate, and actually reduced to acquiring skills in applying algebraic methods to elementary geometry. Exercises throughout the book test the readers understanding of the material and sometimes illustrate extensions of the theory. Curvature, frame fields, and the gaussbonnet theorem. A topological gaussbonnet theorem 387 this alternating sum to be. It should not be relied on when preparing for exams. The gauss bonnet theorem, like few others in geometry, is the source of many fundamental discoveries which are now part of the everyday language of the modern geometer. Orient these surfaces with the normal pointing away from d. Disquisitiones generales circa superficies curvas, 1827. The study of this theorem has a long history dating back to gausss theorema egregium latin. Chapter 3 is an introduction to riemannian geometry.
The gaussbonnet theorem for complete manifolds 749 with a straightforward calculation from 1. The purpose is to extend some wellknown geometric results that hold for the complex torus c. Though this paper presents no original mathematics, it carefully works through the necessary tools for proving gaussbonnet. Gausss divergence theorem let fx,y,z be a vector field continuously differentiable in the solid, s.
In this lecture we introduce the gauss bonnet theorem. Consequences of gaussbonnet one interesting consequence of gaussbonnet is an equation for the area of spherical triangles. Lectures on chernweil theory and witten deformations. The gaussbonnet theorem combines almost everything we have learnt in one. In this article, we shall explain the developments of the gaussbonnet theorem in the last 60 years. The main proof was presented here the paper is behind a paywall, but there is a share link from elsevier, for a few days january 19, 2020. The gaussbonnetchern theorem on riemannian manifolds yin li abstract this expository paper contains a detailed introduction to some important works concerning the gaussbonnetchern theorem. The curvature of the shape is used, as well as its euler characteristic. The gaussbonnet theorem can be seen as a special instance in the theory of characteristic classes. A first course in differential geometry by woodward. Apr 15, 2017 this is the heart of the gaussbonnet theorem. Refer to do carmos proof of the global gaussbonnet theorem 4. The gaussbonnet theorem has also been generalized to riemannian polyhedra.
Aug 07, 2015 here we study the proof of the gauss bonnet theorem based on a rectangularization of a compact oriented surface. About gaussbonnet theorem mathematics stack exchange. This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the nankai institute of mathematics. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and stokes theorem. As wehave a textbook, this lecture note is for guidance and supplement only. Here, we give a simple proof of the general gaussbonnet theorem, essentially. Gaussbonnet theorem simple english wikipedia, the free.
It is also the language used by einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. 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. The study of this theorem has a long history dating back to gauss s theorema egregium latin. S the boundary of s a surface n unit outer normal to the surface. A gaussbonnet theorem, chern classes and an adjunction formula for reductive groups phd thesis valentina kiritchenko department of mathematics university of toronto 2004 abstract my thesis is about geometry of reductive groups. Differential geometry is the study of curved spaces using the techniques of calculus.
This introductory textbook originates from a popular course given to. Gaussbonnet theorem to the o ptical metric, whose geo desics are the spatial ligh t rays, w e found that the fo cusing of light ra ys can b e regar ded as a top ological e. Likewise, the gauss image nb of the entire front face b of the cube is the front pole of s2, and the gauss image nc of the right face c is the east pole of s 2. Latin text and various other information, can be found in dombrowskis book 1. The gaussbonnet theorem in 3d space says that the integral of the gaussian curvature over a closed smooth surface is equal to 2. We present a selfcontained proof of the gaussbonnet theorem for twodimensional. The gauss bonnet theorem bridges the gap between topology and di erential geometry. In this article, we shall explain the developments of the gaussbonnet theorem in. The gaussbonnet theorem department of mathematical. This is a great mathematics book cover the following topics.
Gauss s formulas, christoffel symbols, gauss and codazzimainardi equations, riemann curvature tensor, and a second proof of gauss s theorema egregium. It concerns a surface s with boundary s in euclidean 3space, and expresses a relation between. All the way with gaussbonnet and the sociology of mathematics. Theorem gausss theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. The gauss bonnet theorem links differential geometry with topol ogy. Riemann curvature tensor and gausss formulas revisited in index free notation. As much as it pains me to say it, this proof illustrates the usefulness of indices. The book is, therefore, aimed at professional training of the school or university teachertobe. For a sphere with radius rand a spherical triangle with interior angles 1.
Let us suppose that ee 1 and ee 2 is another orthonormal frame eld computed in another coordinate system u. The theorem tells us that there is a remarkable invariance on. Bonnets theorem on the diameter of an oval surface. It is a nice convenience in the proof of the gaussbonnet formula.
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