Gauss bonnet theorem pdf

See robert greenes notes here, or the wikipedia page on gaussbonnet, or perhaps john lees riemannian manifolds book. 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. The total gaussian curvature of a closed surface depends only on the topology of the surface and is equal to 2. A general gaussbonnet formula that takes into account both formulas can also be given.

The following expository piece presents a proof of this theorem, building up all of the necessary topological tools. Within the proof of the gauss bonnet theorem, one of the fundamental. It is intrinsically beautiful because it relates the curvature of a manifolda geometrical objectwith the its euler characteristica topological one. Historical development of the gaussbonnet theorem article pdf available in science in china series a mathematics 514. The following expository piece presents a proof of this theorem, building.

A fantastic introduction that explains the gauss bonnet theorem in intuitive terms is geometry and topology in manyparticle systems by. Pdf historical development of the gaussbonnet theorem. Millman and parker 1977 give a standard differentialgeometric proof of the gauss bonnet theorem, and singer and thorpe 1996 give a gauss s theorema egregiuminspired proof which is entirely intrinsic, without any reference to the ambient euclidean space. Let s be a closed orientable surface in r 3 with gaussian curvature k and euler characteristic. The gaussbonnet theorem for cone manifolds and volumes of moduli spaces. The gaussbonnet theorem department of mathematical. The gaussbonnet theorem and all of the previously mentioned extensions are speci c instances of this theorem. 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.

A sphere s rp is a subgraph g of x whose vertices are the set of points in g which have geodesic distance r to p normalized so that adjacent points. We introduce the rst and second fundamental forms, central for the study of the local geometry of surfaces. A gaussbonnettype formula on riemannfinsler surfaces with nonconstant indicatrix volume itoh, j. The gaussbonnet theorem that the sum of the interiorangles of a triangle in the plane equals. 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 cherngaussbonnet theorem gives a formula that computes the euler characteristic of an evendimensional smooth manifold as the integration of a curvature characteristic form of the levicivita connection on its tangent bundle. We begin the mathematical part of this paper by looking at the gaussbonnet theorem on the simplest of curved surfaces, the sphere. The gauss bonnet theorem relates the curvature of a surface to a topological property called the euler characteristic. The chern gauss bonnet theorem gives a formula that computes the euler characteristic of an evendimensional smooth manifold as the integration of a curvature characteristic form of the levicivita connection on its tangent bundle. Within the proof of the gaussbonnet theorem, one of the fundamental. In our proof of the curvature 12 theorem, the concept of dimension for abstract graphs plays an important role.

The gaussbonnet theorem is an important theorem in differential geometry. The gaussbonnet theorem the gaussbonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. See robert greenes notes here, or the wikipedia page on gauss bonnet, or perhaps john lees riemannian manifolds book. 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. An examination of the gaussbonnet integrand at one point of m leads one to an extremely difficult algebraic problem which has been resolved in dimension 4 by j. This theorem relates curvature geometry to euler characteristic topology. 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. Theorem gausss theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. It should not be relied on when preparing for exams.

In particular, recall that the heat kernel proof of the. The gaussbonnet theorem for cone manifolds and volumes of moduli spaces the harvard community has made this article openly available. 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. Cherngaussbonnet theorem for graphs pdf, on arxiv nov 2011 and.

Gauss bonnet theorem to prove the poincar ehopf theorem. The gaussbonnet theorem for cone manifolds and volumes. 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 is a special case when m is a 2d manifold. For this, a short introduction to surfaces, di erential forms and vector analysis is given. No matter which choices of coordinates or frame elds are used to compute it, the gaussian curvature is the same function. This proof can be found in guillemin and pollack 1974. Part xxi the gauss bonnet theorem the goal for this part is to state and prove a version of the gauss bonnet theorem, also known as descartes angle defect formula. Bonnet theorem, which asserts that the total gaussian curvature of a compact oriented 2dimensional riemannian manifold is independent of the riemannian metric. In short, it is a 2manifold with or without boundary which is equipped with a riemannian metric.

Papers and updates harvard department of mathematics. 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. Introduction to differential geometry 4 the global gaussbonnet theorem is a truly remarkable theorem. The gaussbonnet theorem combines almost everything we have learnt in. A fantastic introduction that explains the gaussbonnet theorem in intuitive terms is geometry and topology in manyparticle systems by. Consider a surface patch r, bounded by a set of m curves. The gauss bonnet theorem and all of the previously mentioned extensions are speci c instances of this theorem. The gaussbonnet theorem, or gaussbonnet formula, is an important statement about. We show the euler characteristic is a topological invariant by proving the theorem of the classi cation. We are finally in a position to prove our first major localglobal theorem in riemannian geometry. Integrals add up whats inside them, so this integral represents the total amount of curvature of the manifold. Millman and parker 1977 give a standard differentialgeometric proof of the gaussbonnet theorem, and singer and thorpe 1996 give a gausss theorema egregiuminspired proof which is entirely intrinsic, without any reference to the ambient euclidean space. A historical survey of the gaussbonnet theorem from gauss to chern.

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. Gianmarco molino sigma seminar the gaussbonnet theorem 1 februrary, 2019 2223. For surfaces the theorem simplifies and in this simpler version is the older gauss bonnet theorem. The right hand side is some constant times the 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. Gaussbonnet theorem an overview sciencedirect topics. The study of this theorem has a long history dating back to gauss s theorema egregium latin. 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. A historical survey of the gauss bonnet theorem from gauss to chern. Under which conditions this curvature 60 theorem holds is still under investigation. The gaussbonnet theorem is a profound theorem of differential geometry, linking global and local geometry.

But k depends strongly, at a given point, on the rst fundamental form. The gaussbonnetchern theorem on riemannian manifolds. Which of the shown surfaces can be deformed into each. Important applications of this theorem are discussed. Pdf generalized noether theorem for gaussbonnet cosmology. Differential geometry, gauss bonnet theorem, gaussian curvature, gauss map, geodesic curvature, theorema egregium, euler index, genus of a surface. A general gauss bonnet formula that takes into account both formulas can also be given. The goal of this section is to give an answer to the following.

The gauss bonnet theorem links differential geometry with topol ogy. Millman and parker 1977 give a standard differentialgeometric proof of the gaussbonnet theorem, and. A generalization of the gaussbonnet theorem 143 mann structure. Within the proof of the gaussbonnet theorem, one of the fundamental theorems is applied. Let us suppose that ee 1 and ee 2 is another orthonormal frame eld computed in another coordinate system u. The gauss bonnet theorem is an important theorem in differential geometry. 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 goal of these notes is to give an intrinsic proof of the gau. Theorem gauss s theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. On the dimension and euler characteristic of random graphs pdf. In 1603 harriot1 showed that on a sphere of radius 1 the area of a spherical triangle that is, a. The gauss bonnet theorem, or gauss bonnet 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. For surfaces the theorem simplifies and in this simpler version is the older gaussbonnet theorem.

The gaussbonnet theorem relates the sum of the interior angles of a triangle with the its gaussian curvature, an intrinsic quantity. As wehave a textbook, this lecture note is for guidance and supplement only. 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. 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 relates the curvature of a surface to a topological property called the euler characteristic. Pdf a discrete gaussbonnet type theorem semantic scholar. A topological gaussbonnet theorem 387 this alternating sum to be. Gaussian curvature and the gaussbonnet theorem universiteit. Gianmarco molino sigma seminar the gauss bonnet theorem 1 februrary, 2019 2223. The gaussbonnet theorem for cone manifolds and volumes of. Introduction to differential geometry 4 the global gauss bonnet theorem is a truly remarkable theorem. It concerns a surface s with boundary s in euclidean 3space, and expresses a relation between.

The study of this theorem has a long history dating back to gausss theorema egregium latin. The gauss bonnet theorem links di erential geometry with topology. 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. 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 says that, for a closed 7 manifold. S the boundary of s a surface n unit outer normal to the surface. Bonnet theorem to higher dimensions is a special case of hirzebruchs riemannroch theorem 5 and involves todd classes.

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. In this lecture we introduce the gaussbonnet theorem. The horospherical gaussbonnet type theorem in hyperbolic space izumiya, shyuichi and romero fuster, maria del carmen, journal of the mathematical society of japan, 2006. 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. In this article, we shall explain the developments of the gaussbonnet theorem in. Rather, it is an intrinsic statement about abstract riemannian 2manifolds. The idea of proof we present is essentially due to. Proofs of the cauchyschwartz inequality, heineborel and invariance of domain theorems. The left hand side is the integral of the gaussian curvature over the manifold. It was remarkable that k is an invariant of local isometries, when the principal curvatures are not. A compact and oriented riemannian manifold of dimension 4.