Characteristic Classes
In 1827 Gauss published his classic book Disquisitiones generales circa superficies curvas. He defined the total curvature (now called the Gaussian curvature) \(\kappa \) as a function on the surface. In his famous theorema egregium Gauss proved that the
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Characteristic Classes
5.1 Introduction In 1827 Gauss published his classic book Disquisitiones generales circa superficies curvas. He defined the total curvature (now called the Gaussian curvature) κ as a function on the surface. In his famous theorema egregium Gauss proved that the total curvature κ of a surface S depends only on the first fundamental form (i.e., the metric) of S.Gauss defined the integral curvature κ(Σ) of a bounded surface Σ to be Σ κ dσ. He computed κ(Σ) when Σ is a geodesic triangle to prove his celebrated theorem κ(Σ) := κ dσ = A + B + C − π, (5.1) Σ
where A, B, C are the angles of the geodesic triangle Σ. Gauss was aware of the significance of equation (5.1) in the investigation of the Euclidean parallel postulate (see Appendix B for more information). He was interested in surfaces of constant curvature and mentions a surface of revolution of constant negative curvature, namely, a pseudosphere. The geometry of the pseudosphere turns out to be the non-Euclidean geometry of Lobaˇcevski– Bolyai. Equation (5.1) is a special case of the well-known Gauss–Bonnet theorem. When applied to a compact, connected surface Σ the Gauss-Bonnet theorem states that κ dσ = 2πχ(Σ), (5.2) Σ
where χ(Σ) is the Euler characteristic of Σ. The left hand side of equation (5.2) is arrived at through using the differential structure of Σ, while the right hand side depends only on the topology of Σ. Thus, equation (5.2) is a relation between geometric (or analytic) and topological invariants of Σ. This result admits far-reaching generalizations. In particular, it can be regarded as a prototype of an index theorem. The Gauss–Bonnet theorem was K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 5, c Springer-Verlag London Limited 2010
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5 Characteristic Classes
generalized to Riemannian polyhedra by Allendoerffer and Weil and to arbitrary manifolds by Chern. In this latter generalization the Gaussian integral curvature is replaced by an invariant formed from the Riemann curvature. It forms the starting point of the theory of characteristic classes. Gauss’ idea of studying the geometry of a surface intrinsically, without leaving it (i.e., by means of measurements made on the surface itself), is of fundamental importance in modern differential geometry and its applications to physical theories. We are similarly compelled to study the geometry of the three-dimensional physical world by the intrinsic method, i.e., without leaving it. This idea was already implicit in Riemann’s work, which extended Gauss’ intrinsic method to the study of manifolds of arbitrary dimension. This work together with the work of Ricci and Levi-Civita provided the foundation for Einstein’s theory of general relativity. The constructions discussed in this chapter extend these ideas and provide important tools for modern mathematical physics.
5.2 Classifying Spaces Let G be a Lie group. The classification of principal G-bundles over a manifold M is achieved by the use of classifying spaces. A topological space Bk (G)
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