Characteristic Classes

Kishore Marathe ()
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Kishore Marathe: City University of New York Brooklyn College

Chapter Chapter 5 in Topics in Physical Mathematics, 2010, pp 137-167 from Springer

Abstract: Abstract 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 total curvature $$\kappa $$ of a surface S depends only on the first fundamental form (i.e., the metric) of S. Gauss defined the integral curvature $$\kappa (\Sigma )$$ of a bounded surface Σ to be $${\int\nolimits \nolimits }_{\Sigma }\kappa \ d\sigma $$ . He computed $$\kappa (\Sigma )$$ when Σ is a geodesic triangle to prove his celebrated theorem 5.1 $$\kappa (\Sigma ) :={ \int\nolimits \nolimits }_{\Sigma }\kappa \ d\sigma= A + B + C - \pi, $$ 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 Lobačevski–Bolyai.

Keywords: Vector Bundle; Cohomology Class; Chern Class; Principal Bundle; Index Theorem (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-1-84882-939-8_5

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