Furthermore, the new composite path and the initial path form a triangular region, called geodesic triangle, which satisfies the local

Gauss-Bonnet Theorem.

The classical

Gauss-Bonnet Theorem implies that the only compact Riemann surface with positive curvature is the Riemann sphere.

Two new sections look at a recent development concerning the

Gauss-Bonnet theorem and scalar curvature for curved noncommutative tori, and Hopf cyclic cohomology.

Chapter 5 studies the intrinsic geometry of n-dimensional submanifolds in pseudo-Euclidean m-dimensional space and Chapter 6 establishes the required machinery and gives the proof of the

Gauss-Bonnet theorem.

Hence, in the compact case, equation (2) reduces to[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is the well known

Gauss-Bonnet theorem for a Riemannian metric on the torus [T.

Topics increase in mathematical complexity as the book progresses to encompass dozens of examples including a derivation of the Euler-Lagrange equation, heat flow and analytic functions, and a bicycle wheel and the

Gauss-Bonnet theorem.

Consequently, the contents are accessible to a wider audience and can be used to prepare students for the study of the Divergence Theorem, Green's Theorem, and even

Gauss-Bonnet Theorem in Differential Geometry.

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.

The crucial fact here is the

Gauss-Bonnet theorem 1.

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.

We apply the

Gauss-Bonnet theorem to the triangle [[delta],bar above] to yield (note the Gaussian curvature K < 0),