The Generalized Gauss-Bonnet Theorem

Abstract

In this chapter we shall prove the Generalized Gauss-Bonnet Theorem using tubes. The principal ingredients are: (1) H. Hopf’s generalization [Hopfl], [Hopf2] of the Gauss-Bonnet Theorem for hypersurfaces in ℝn, (2) the Nash Embedding Theorem [Nash], (3) Weyl’s Tube Formula [Weyll], and (4) some elementary calculations with volumes of tubes and Euler characteristics. This proof using tubes is neither the most direct nor the most elegant; Chern’s proof [Chernl] excels in both of these respects (see the end of this chapter). However, the proof via steps (1)—(4) has many interesting features; it is due to Allendoerfer [All] and Fenchel [F12]. Furthermore, some of the techniques of the present chapter will be useful when we discuss complex versions of the Weyl Tube Formula in Chapters 6 and 7.