Though this paper presents no original mathematics, it carefully works through the necessary tools for proving gaussbonnet. 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. The theorems of greenstokes,gaussbonnet and poincarehopf in graph theory pdf. Use the gaussbonnet theorem to prove that circles on the unit sphere that are not big circles are not geodesics. About gaussbonnet theorem mathematics stack exchange. The gaussbonnet theorem relates the sum of the interior angles of a triangle with the its gaussian curvature, an intrinsic quantity of the geometry. Already one can see the connection between local and global geometry. Let abc be a geodesic triangle on a smooth oriented surface x in euclidean 3space with angles. It should not be relied on when preparing for exams. Integrals add up whats inside them, so this integral represents the total amount of curvature of the manifold. The next theorem asserts that r c rfdr fb fa, where fis a function of two or three variables and cis. 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.
Thus, a surface of genus 2 is never periodic, and a minimal surface of genus g in a 3torus t 3 has 4 g. Here s1p is the arc length of the unit sphere of p and s2p is the arc length of the sphere of. Part xxi the gaussbonnet theorem the goal for this part is to state and prove a version of the gaussbonnet theorem, also known as descartes angle defect formula. Whether or not the compact surface admits an isometric or even just differentiable embedding in plays no role in the statement. Prove that for anyp2swithpthe corresponding point on the plane, there holds gp sp. Bonnet is interpreted within the framework of complex analysis of one and several variables.
In short, it is a 2manifold with or without boundary which is equipped with a riemannian. The simplest case of gb is that the sum of the angles in a planar triangle is 180 degrees. Jm jdm here p is the section of sm over dm given by the outward unit normal vector. Pdf a discrete gaussbonnet type theorem semantic scholar. Then well state and explain the gaussbonnet theorem and derive a number of consequences. The gaussbonnet theorem in this lecture we will prove two important global theorems about the geometry and topology of twodimensional manifolds.
We are finally in a position to prove our first major localglobal theorem in riemannian geometry. 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. Looking forward to a detailed explanation or references on this particular explanation. Rather, it is an intrinsic statement about abstract riemannian 2manifolds. Gaussbonnet theorem related the topology of a manifold to its geometry. The next theorem asserts that r c rfdr fb fa, where fis a function of two or three variables and cis a curve from ato b.
This theorem relates curvature geometry to euler characteristic topology. The idea of proof we present is essentially due to. In order to prove the jordan curve theorem, we list several lemmas without proof. The classical gaussbonnet theorem expresses the curvatura integra, that is, the integral of the gaussian curvature, of a curved polygon in terms of the angles of the polygon and of the geodesic curvatures. The gaussbonnet theorem department of mathematical. The rst equality is the gaussbonnet theorem, the second is the poincar ehopf index theorem. To prove this we pass from the manifold r to the manifold m2, of 2n 1 dimensions formed by the unit. In this lecture we introduce the gaussbonnet theorem. Special relativity, electrodynamics and general relativity.
But the gaussbonnet theorem is particularly easy to prove in case the surface is isometrically embedded in. These notions of curvature tell us roughly what a surface looks like both locally and globally. Introduction the purpose of this paper is to prove that every di erential character 11, 3 can be represented by di erential form with singularities and to prove a version of the gaussbonnetchern theorem gbc theorem for vector bundles taking values in di erential characters. Within the proof of the gaussbonnet theorem, one of the fundamental theorems is applied. The total gaussian curvature of a closed surface depends only on the topology of the surface and is equal to 2. As wehave a textbook, this lecture note is for guidance and supplement only.
Gaussian curvature and the gaussbonnet theorem universiteit. See robert greenes notes here, or the wikipedia page on gauss bonnet, or perhaps john lees riemannian manifolds book. 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. 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. We derive the gauss bonnet theorem in the framework of classical differential geometry. Cherngaussbonnet theorem for graphs pdf, on arxiv nov 2011 and. A lift of the gaussbonnetchern theorem 8 references 10 1. 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. I was wondering if there was a prove of the poincarehopf index theorem using gaussbonnet ill really like to see one to understand it better. Next, we develop integration and cauchys theorem in various guises, then apply this to the study of analyticity, and harmonicity, the logarithm and the winding number. In this article, we shall explain the developments of the gaussbonnet theorem in. Bonnet theorem, which asserts that the total gaussian curvature of a compact oriented 2dimensional riemannian manifold is independent of the riemannian metric. Gauss theorem 3 this result is precisely what is called gauss theorem in r2.
See robert greenes notes here, or the wikipedia page on gaussbonnet, or perhaps john lees riemannian manifolds book. At that point we have all the ingredients to prove the gauss bonnet theorem for manifolds without boundary. The sides of the triangle are geodesics, that is, arcs of great circles. A graph theoretical gaussbonnetchern theorem oliver knill abstract.
Pdf we derive the gauss bonnet theorem in the framework of classical differential geometry. On the cherngaussbonnet theorem for the noncommutative 4. Its importance lies in relating geometrical information of a surface to a purely topological characteristic, which has resulted in varied and powerful applications. I have just started reading on gaussbonnet theorem and i guess my query lies at the heart of the underlying philosophy of this gem theorem of differential topology. All concept and numericals jee mainsneet ii duration. Let s be a closed orientable surface in r 3 with gaussian curvature k and euler characteristic. There are other physicalexplanations for gaussbonnet, for exampleseehere. State and prove the gaussbonnet theorem for a spherical polygon with geodesic sides. Gaussian curvature and its independence of coordinates. The integrand in the integral over r is a special function associated with a vector. Our special emphasis is the relation of this theorem to different areas including characteristic classes, probability, polyhedra and physics. Suppose mis a compact oriented 2dimensional manifold, and assume.
Invariance of gaussbonnet theorem with respect to connection. The gaussbonnet theorem is an important theorem in differential geometry. The gaussbonnet theorem for complete manifolds 747 now suppose m is incomplete with boundary dm. A more detailed version can be found ina bicycle wheel proof of the gaussbonnet theorem, mark levi, expo. Expert answer 100% 2 ratings previous question next question get more help from chegg. 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. We conclude the chapter with some brief comments about cohomology and the fundamental group.
The idea of proof we present is essentially due to s. In section 4, we prove the gaussbonnet theorem for compact surfaces by considering triangulations. A topological gaussbonnet theorem 387 this alternating sum to be. The theorem tells us that there is a remarkable invariance on. We prove a prototype curvature theorem for subgraphs g of the flat triangular tesselation which play the analogue of domains in two dimensional euclidean space. We show the euler characteristic is a topological invariant by proving the theorem of the classi cation. Hopfs generalization hopfl, hopf2 of the gaussbonnet theorem for hypersurfaces in. A concise course in complex analysis and riemann surfaces. At that point we have all the ingredients to prove the gaussbonnet theorem for manifolds without boundary. The gaussbonnet theorem says that, for a closed 7 manifold. The pusieux curvature kp 2s1p s2p is equal to 12 times the euler characteristic when summed over the boundary of g. The left hand side is the integral of the gaussian curvature over the manifold. It is an extraordinary result which expresses the total gaussian curvature of a compact manifold in terms of its euler characteristic a topological invariant.
What is the significance of the gaussbonnet theorem. Gaussbonnet theorem an overview sciencedirect topics. Gauss bonnet theorem u of u math university of utah. The gauss bonnet theorem bridges the gap between topology and di erential geometry.
On the dimension and euler characteristic of random graphs pdf. The sum of the angles of a triangle is equal to equivalently, in the triangle represented in figure 3, we have. Orient these surfaces with the normal pointing away from d. In this article, we shall explain the developments of the gaussbonnet theorem in the last 60 years. In this paper we survey some developments and new results on the proof and applications of the gaussbonnet theorem. It is intrinsically beautiful because it relates the curvature of a manifolda geometrical objectwith the its euler characteristica topological one. The gaussbonnet theorem the gaussbonnet theorem is one of the most beautiful. A region bounded by a simple closed curved with three vertices and three edges is called a triangle. Pdf supplemental lecture 2 derivation of the gauss bonnet. Next well try to understand on intuitive grounds why the gaussbonnet theorem is true. In chapter 6 we will nally prove the gauss bonnet theorem. The far reaching significance of the theorem is discussed.
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 gaussbonnet theorem is a special case when m is a 2dimensional manifold. In chapter 6 we will nally prove the gaussbonnet theorem. Since it is a topdimensional differential form, it is closed. Note that if p n 2k, then integrating each side of 7 over m gives theorem 1. The gaussbonnetchern theorem is obtained from theorem 1 by taking e to be the tangent bundle of an orientable riemannian manifold m, endowed with the levicivita connection. The gauss bonnet theorem, or gauss bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. Chapter 18 the theorems of green, stokes, and gauss. Feb 07, 2018 electric charges and fields gauss law. 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 can be seen as a special instance in the theory of characteristic classes.
The formula of gauss bonnet in question asserts that the integral of this differential form over r is equal to the eulerpoincar6 characteristic x of r. In particular, we prove the gaussbonnet theorem in that case. Apr 15, 2017 this is the heart of the gaussbonnet theorem. 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. The naturality of the euler class means that when changing the riemannian metric, one stays in the same cohomology class.
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