Local coordinates are necessary, but coordinatefree concepts are emphasized. The gaussbonnet theorem is a profound theorem of differential geometry, linking global and local geometry. 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 gauss bonnet integrand as with the twodimensional gauss bonnet theorem, there are generalizations when m is a manifold with boundary. This formula can of course also be used to prove the global result, but bonnet doesnt seem to have observed this either. A topological gaussbonnet theorem 387 this alternating sum to be. This proof can be found in guillemin and pollack 1974. We introduced the gauss bonnet theorem in chapter 12 of the textbook and applied it to problems concerning the local and global properties of surfaces. It should not be relied on when preparing for exams. The rst equality is the gaussbonnet theorem, the second is the poincar ehopf index theorem.
The gauss bonnet theorem, or gauss bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. Then we extend the local gauss bonnet theorem to the global one. Consider a surface patch r, bounded by a set of m curves. Within the proof of the gaussbonnet theorem, one of the fundamental theorems is. Its one of my favourite theorems and is probably the best one to work towards if you dont know any dg. The left hand side is the integral of the gaussian curvature over the manifold. The theorems of greenstokes, gauss bonnet and poincarehopf in graph theory pdf. Zf the set of spherical leaves ifpnegligible, then mean value of the curvature, i. The total gaussian curvature of a closed surface depends only on the topology of the surface and is equal to 2. The gaussbonnetchern theorem on riemannian manifolds.
Gaussbonnet theorem an overview sciencedirect topics. Extensions of the gauss bonnet theorem to more general open surfaces have been investigated in the past. The gauss bonnet theorem demonstrates how integral of curvature, a geometric property, corresponds with euler characteristic, a topological measurement. This is a local global theorem par excellence, because it asserts the equality of two very differently defined quantities on a compact, orientable riemannian 2manifold m. The levicivita connection is presented, geodesics introduced, the jacobi operator is discussed, and the gauss bonnet theorem is proved. In this lecture we introduce the gauss bonnet theorem.
Aspects of differential geometry i synthesis lectures on. You cut it up into geodesic triangles, you apply the angle excess theorem to each of those triangles, you add them all up, and you count very carefully based on the graph. His simple intrinsic proof in 1944 not only gave a beautiful and profound proof of the gauss bonnet theorem, but also enlightened the whole. An example of a complex region where gaussbonnet theorem can apply. We now state the global gauss bonnet theorem for surfaces without boundaries. The global gaussbonnet theorem is a truly remarkable theorem. The gaussbonnet formula is a beautiful example of a localtoglobal result. The chern gauss bonnet theorem is usually viewed as a way of relating the curvature of the manifold with. Historical development of the gaussbonnet theorem article pdf available in science in china series a mathematics 514.
As wehave a textbook, this lecture note is for guidance and supplement only. Pdf a historical survey of the gaussbonnet theorem from gauss to chern. Integrals add up whats inside them, so this integral represents the total amount of curvature of the manifold. In this note we give a proof of the gaussbonnet theorem for riemannian manifolds of any dimension using morse theory.
Millman and parker 1977 give a standard differentialgeometric proof of the gaussbonnet theorem, and. Pdf we derive the gauss bonnet theorem in the framework of. The lecture notes i am using however does not have the integral with the geodesic curvature but they seem to have the same hypotheses. We assume that the surface is bounded, of finite area and without boundary. For m a compact orientable surface, it states that. The gauss bonnet theorem is a special case when m is a 2d manifold. The gaussbonnet theorem department of mathematical.
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. In section 4, we prove the gaussbonnet theorem for compact surfaces by considering triangulations. It concerns a surface s with boundary s in euclidean 3space, and expresses a relation between. The gauss bonnet theorem links differential geometry with topol ogy. We show the euler characteristic is a topological invariant by proving the theorem of the classi cation. 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. The fact that the integral of the gaussian curvature conspires to be a purely topological. Local calculations relate to global topological invariants, as exemplified by the gauss bonnet theorem. Gauss bonnet formula in the spirit of the gackstatterjorgemeeks formula 3, 4 for minimal surfaces. On the dimension and euler characteristic of random graphs pdf. I will then discuss gaussian curvature and geodesics. If the triangle is in the poincaire model of noneuclidean geometry. We have proven the local case of this theorem, and the global theorem. Gaussian curvature and the gaussbonnet theorem universiteit.
The gaussbonnet theorem 183 2 1 0 1 2 y 2 1 1 2 x vx2. 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. A summary of topological facts about surfaces is given in section 27. In topology, is said to be compact without boundary. The following expository piece presents a proof of this theorem, building up all of the necessary topological tools. In the late 1970s, orbifolds were use by thurston in his geometrization program for threemanifolds.
Let us suppose that ee 1 and ee 2 is another orthonormal frame eld computed in another coordinate system u. 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. We begin by subdividing the given region r into many small triangles, each of which fits inside the. The global gauss bonnet theorem is a truly remarkable theorem. The following expository piece presents a proof of this theorem, building. In fact, as far as we can discover, the rst proof of the global gauss bonnet theorem for embedded surfaces. 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. For surfaces the theorem simplifies and in this simpler version is the older gauss bonnet theorem. It was thurston who changed the name from vmanifold to orbifold. Differential geometry in graphs harvard university. Suppose mis a compact oriented 2dimensional manifold, and assume. We are finally in a position to prove our first major local global theorem in riemannian geometry. The overall idea behind local global principles is that you piece together local information into some sort of global picture. Theorem 3 suppose a triangle has geodesic sides and angles.
As an application of his foliated index theorem, connes proved the following gauss bonnet type theorem. Math 501 differential geometry herman gluck thursday march 29, 2012. Already one can see the connection between local and global geometry. The material is appropriate for an undergraduate course in the. In 1985, with the work of dixon, harvey, vafa and witten on conformal eld theory 7, the.
Episode 11 jeanne nielsen clelland january 24, 2018 kevin knudson. For this, a short introduction to surfaces, di erential forms and vector analysis is given. Pdf supplemental lecture 2 derivation of the gauss bonnet. The gaussbonnet theorem, or gaussbonnet formula, is an important statement about. The gaussbonnet theorem the gaussbonnet theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. Pdf historical development of the gaussbonnet theorem. Differential geometry, gauss bonnet theorem, gaussian curvature, gauss map, geodesic curvature, theorema egregium, euler index, genus of a surface. Within the proof of the gauss bonnet theorem, one of the fundamental.
The idea of proof we present is essentially due to. The right hand side is some constant times the euler characteristic. The gaussbonnet theorem in this lecture we will prove two important global theorems about the geometry and topology of twodimensional manifolds. Gianmarco molino sigma seminar the gauss bonnet theorem 1 februrary, 2019 1623. It was chern who made the gauss bonnet theorem 6 widely known. 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 simplest case of gb is that the sum of the angles in a planar triangle is 180 degrees. The gauss bonnet theorem comes in local and global version. Theorem gauss s theorema egregium, 1826 gauss curvature is an invariant of the riemannan metric on. Curvature and gaussbonnet defect of global affine hypersurfaces. It was remarkable that k is an invariant of local isometries, when the principal curvatures are not. The gauss bonnet theorem links di erential geometry with topology. That theorem is gauss bonnet, which links global topological properties of a space with the local property of curvature. 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. Pdf curvature and gaussbonnet defect of global affine. The gauss bonnet with a t at the end theorem is one of the most important theorem in the differential geometry of surfaces. Here d is the exterior derivative, and d is its formal adjoint under the riemannian metric. A similar statement holds also for codimension two surfaces see theorem 3. A region bounded by a simple closed curved with three vertices and three edges is called a triangle. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and stokes theorem.
Chapter 3 is an introduction to riemannian geometry. No matter which choices of coordinates or frame elds are used to compute it, the gaussian curvature is the same function. The gaussbonnet theorem says that, for a closed 7 manifold. But k depends strongly, at a given point, on the rst fundamental form. 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. Important applications of this theorem are discussed.
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