The gauss bonnet theorem is a formula that yields a topological invariant, i. 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. The method canm of course be applied to derive other formulas of the same type and, with suitable modifications, to deduce the gauss bonnet formula for. This is kind of weird, since you wouldnt normally think of a donut and a coffee mug as the same shape. Historical development of the gaussbonnet theorem springerlink. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. The gauss bonnet theorem, or gauss bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. These notions of curvature tell us roughly what a surface looks like both locally and globally. Apr 15, 2017 this is the heart of the gaussbonnet theorem. In short, it is a 2manifold with or without boundary which is equipped with a riemannian metric. The gauss bonnet theorem bridges the gap between topology and di erential geometry. Mountains, earthquakes, and the gauss bonnet theorem the gauss bonnet 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.
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. It relates a compact surfaces total gaussian curvature to its euler characteristic. Let us suppose that ee 1 and ee 2 is another orthonormal frame eld computed in another coordinate system u. It should not be relied on when preparing for exams. See robert greenes notes here, or the wikipedia page on gaussbonnet, or perhaps john lees riemannian manifolds book. A more detailed version can be found ina bicycle wheel proof of the gauss bonnet theorem, mark levi, expo.
The goal of these notes is to give an intrinsic proof of the gau. A few new topics have been added, notably sards theorem and transversality, a proof that infinitesimal lie group actions generate global group actions, a more. The gaussbonnet theorem is a special case when m is a 2dimensional manifold. The gauss bonnet theorem comes in local and global version. No matter which choices of coordinates or frame elds are used to compute it, the gaussian curvature is the same function.
The following expository piece presents a proof of this theorem, building. For surfaces the theorem simplifies and in this simpler version is the older gauss bonnet theorem. The idea of proof we present is essentially due to. Aug 07, 2015 here we study the proof of the gauss bonnet theorem based on a rectangularization of a compact oriented surface. The gaussbonnet theorem, essentially the point toward which this entire book has been aimed, shows that these two approaches are deeply related. Let s be a closed orientable surface in r 3 with gaussian curvature k and euler characteristic. The gauss bonnet theorem implies that if m g is a minimal surface of genus g in a 3tours t 3, then its gauss map g. See robert greenes notes here, or the wikipedia page on gauss bonnet, or perhaps john lees riemannian manifolds book. The book begins with a nonrigorous overview of the subject in chapter 1, designed to introduce some of the intuitions underlying the notion of curvature and to link them with elementary geometric ideas the student has seen before. The gaussbonnet theorem the gaussbonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. The gaussian curvature can also be negative, as in the case of a. Finally, an application to physics of a corollary of the gauss bonnet theorem is presented involving the behaviour of liquid crystals on a spherical shell.
The gauss bonnet theorem is a theorem that connects the geometry of a shape with its topology. A fantastic introduction that explains the gauss bonnet theorem in intuitive terms is geometry and topology in manyparticle systems by. Differential geometry in graphs harvard university. 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 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. This course is about a profound and far reaching generalization of the gauss bonnet theorem, or perhaps given how farreaching the generalization is, it might better be said that the gauss bonnet theorem is the simplest nontrivial instance of the result the atiyahsinger index theorem we are aiming at. Apart from being interesting in their own right, these discrete concepts might. Moscovici, the l2index theorem for homogeneous spaces of lie groups, ann. The latter requires both a notion of distance and differentiability. The gauss bonnet with a t at the end theorem is one of the most important theorem in the differential geometry of surfaces. The author also gives a very brief introduction to noneuclidean geometry. Theorem gausss theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. The theorem tells us that there is a remarkable invariance on.
The gaussbonnet theorem is a theorem that connects the geometry of a shape with its topology. R n, the sum of whose indices is, by a wellknown theorem, equal to x. The simplest one expresses the total gaussian curvature of an embedded triangle in terms of the total. As wehave a textbook, this lecture note is for guidance and supplement only. Jan 25, 2018 the gauss bonnet theorem states that the total curvature of such a surface, or the integral of the curvature over the service, depends only on the number of holes it has, also known as its genus. It is intrinsically beautiful because it relates the curvature of a manifolda geometrical objectwith the its euler characteristica topological one. Proofs of the cauchyschwartz inequality, heineborel and invariance of domain theorems. The gaussbonnet theorem can be seen as a special instance in the theory of characteristic classes. The gaussbonnet theorem and the surface of revolution curvature theorem says the total curvature for the lens is equal to 4 the integral of curvature on the smooth surfaces of the lens can be evaluated separately so the contribution of the sharp ridge can be found as the difference. We develop some preliminary di erential geometry in order to state and prove the gauss bonnet theorem, which relates a compact surfaces gaussian curvature to its euler characteristic. This is a history of the gaussbonnet theorem as i see it. The gaussbonnet theorem states that the total curvature of a closed twodimensional oriented surface i.
The total gaussian curvature of a closed surface depends only on the topology of the surface and is equal to 2. 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. The gaussbonnet theorem is one of the most beautiful and one of the deepest. Gaussbonnet theorem an overview sciencedirect topics. The naturality of the euler class means that when changing the riemannian metric, one stays in the same cohomology class. The gauss bonnet theorem, essentially the point toward which this entire book has been aimed, shows that these two approaches are deeply related. The fundamental theorem of algebra states that every nonconstant singlevariable polynomial with complex coefficients has at least one complex root. 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. We show the euler characteristic is a topological invariant by proving the theorem of the classi cation. Latin text and various other information, can be found in dombrowskis book 1.
The book ends with a thorough treatment of the gauss bonnet theorem for smooth surfaces. For example, a sphere of radius r has gaussian curvature 1 r 2 everywhere, and a flat plane and a cylinder have gaussian curvature zero everywhere. For m a compact orientable surface, it states that. Pdf historical development of the gaussbonnet theorem. Within the proof of the gauss bonnet theorem, one of the fundamental theorems is applied. Its importance lies in relating geometrical information of a surface to a purely topological characteristic, which has resulted in varied and powerful applications. Its importance lies in relating geometrical information of a surface to a purely topological characteristic, which has. 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. The gauss bonnet theorem will be a recurring theme in this book and we will provide several other proofs and generalizations. The gaussbonnet theorem implies that if m g is a minimal surface of genus g in a 3tours t 3, then its gauss map g.
Chern gauss bonnet theorem for graphs pdf, on arxiv nov 2011 and updates. Historical development of the gaussbonnet theorem article pdf available in science in china series a mathematics 514. Jeanne clellands favorite theorem scientific american blog. Thus, a surface of genus 2 is never periodic, and a minimal surface of genus g in a 3torus t 3 has 4 g. In particular, if t is not at everywhere, then it contains elliptic, parabolic and at points. The gaussbonnet theorem is obviously not at the beginning of the. It concerns a surface s with boundary s in euclidean 3space, and expresses a relation between.
Part of the graduate texts in mathematics book series gtm, volume 176. Looking forward to a detailed explanation or references on this particular explanation. There are other physicalexplanations for gauss bonnet, for exampleseehere. The gauss bonnet theorem in 3d space says that the integral of the gaussian curvature over a closed smooth surface is equal to 2.
The gaussbonnet theorem is an important theorem in differential geometry. According to marcel berger in his book, a panoramic view of riemannian geometry, this formula for the area of a spherical triangle was discovered by thomas harriot 15601621 in 1603. Several results from topology are stated without proof, but we establish almost all. There is evidence that descartes knew about this formula a century before euler, s. The gauss bonnet theorem relates the curvature of a surface to a topological property called the euler characteristic. Jan, 2010 characteristic, and it is immediate to prove a discrete gauss bonnet theorem see theorem 3. About gaussbonnet theorem mathematics stack exchange. In differential geometry, the gaussian curvature or gauss curvature. What is the significance of the gaussbonnet theorem. The gauss bonnet theorem is even more remarkable than the theorema egregium.
Integrals add up whats inside them, so this integral represents the total amount of curvature of the manifold. The study of this theorem has a long history dating back to gauss s theorema egregium latin. This is not a theorem about minimal surfaces, but it is probably the most important theorem in surface theory, and it plays a role in projects 2 and 5 and is relevant to chapter 9 of osserman, which is the last section we will cover in this course. We are finally in a position to prove our first major localglobal theorem in riemannian geometry. The curvature of the shape is used, as well as its euler characteristic. The gaussbonnet theorem is a profound theorem of differential geometry, linking global and local geometry. Theorem 3 suppose a triangle has geodesic sides and angles. Bonnet s theorem on the diameter of an oval surface. It is a vast generalization of a formula involving convex polyhedra due to euler.
Gaussbonnet theorem article about gaussbonnet theorem by. The gauss image of the common edge shared by the faces. The gaussbonnet theorem says that, for a closed 7 manifold. Review of basics of euclidean geometry and topology. Gauss bonnet theorem related the topology of a manifold to its geometry.
With such an interpretation the integral of hi over the boundary of vn can be evaluated and is easily proved to be equal to x. In this article, we shall explain the developments of the gaussbonnet theorem in the last 60 years. In this notation we can write the gaussbonnet theorem as the gaussbonnethopf theorem. The gaussbonnet theorem, or gaussbonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their. Though this paper presents no original mathematics, it carefully works through the necessary tools for proving gaussbonnet. On the dimension and euler characteristic of random graphs pdf. Furthermore, the new composite path and the initial path form a triangular region, called geodesic triangle, which satisfies the local gauss bonnet theorem. I have just started reading on gauss bonnet theorem and i guess my query lies at the heart of the underlying philosophy of this gem theorem of differential topology. Gaussbonnet theorem simple english wikipedia, the free. 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. In this lecture we introduce the gauss bonnet theorem. The gauss bonnet theorem the gauss bonnet theorem is one of the most beautiful. Bonnet theorem, which asserts that the total gaussian curvature of a compact oriented 2dimensional riemannian manifold is independent of the riemannian metric. The gaussbonnet theorem topology is the study of shapes and, in particular, what doesnt change when you bend and squish them.
Introduction the gaussbonnet theorem is perhaps one of the deepest theorems of di erential geometry. Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. In this lecture we introduce the gaussbonnet theorem. The euler characteristic is a purely topological property, whereas the gaussian curvature is purely geometric. Rather, it is an intrinsic statement about abstract riemannian 2manifolds. The eulerpoincar e number is the earliest invariant of algebraic topology. A first course in geometric topology and differential.
We develop some preliminary di erential geometry in order to state and prove the gaussbonnet theorem, which relates a compact surfaces gaussian curvature to its euler characteristic. The gaussbonnet theorem is even more remarkable than the theorema egregium. It is an extraordinary result which expresses the total gaussian curvature of a compact manifold in terms of its euler characteristic a topological invariant. Deriving gauss bonnet gravity or just higher order corrections 19. Already one can see the connection between local and global geometry. The left hand side is the integral of the gaussian curvature over the manifold.
The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. I am trying to find that fact but the most i found was a bit complex explanation in the hawking ellis book. I just read in the zees book quantum field theory in a nutshell. The simplest case of gb is that the sum of the angles in a planar triangle is 180 degrees. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. It is named after the two mathematicians carl friedrich gau. The gaussbonnetchern theorem on riemannian manifolds. Since it is a topdimensional differential form, it is closed. So the gauss image na of the entire face a is the north pole of s 2. 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. The gauss bonnet theorem states that the total curvature of a closed twodimensional oriented surface i. As we have a textbook, this lecture note is for guidance and supplement only.
Exercises throughout the book test the readers understanding of the material and sometimes illustrate extensions of the theory. All the way with gaussbonnet and the sociology of mathematics. 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. The gaussbonnet theorem, like few others in geometry, is the source of many. Mathematics volume 51, pages 777 784 2008 cite this article. Singularities of vector fields 15 acknowledgements 18 references 19 1. Historical development of the gaussbonnet theorem hunghsi wu 1 science in china series a. The gauss bonnet theorem links differential geometry with topol ogy.
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