Two new sections look at a recent development concerning the Gauss-Bonnet theorem
and scalar curvature for curved noncommutative tori, and Hopf cyclic cohomology.
Along the way he covers the geometry of curves, surfaces, curvatures, constant mean curvature surfaces, geodesics, metrics, isometries, holonomy and the Gauss-Bonnet theorem
, the calculations of variations and geometry, and higher dimensions, just for fun.
This is the famous Gauss-Bonnet theorem
for compact surface.
Among the topics are nearby cycles and periodicy in cyclic homology, the Gauss-Bonnet theorem
for the noncommutative two torus, zeta phenomenology, absolute modular forms, the transcendence of values of transcendental functions at algebraic points, and the Hopf algebraic structure of perturbative quantum gauge theories.
Topics include the Gauss map and the second fundamental form, the divergence theorem, global extrinsic geometry, rigid motions and isometrics, and the Gauss-Bonnet theorem