MORSE THEORY AND THE GAUSS-BONNET FORMULA Alina ...

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A special thanks to Professor Christopher Croke for all his help. ii
1 Introduction The simplest version of the Gauss-Bonnet theorem was probably known in the time of Thales and states that the sum of the interior angles of a triangle in the Euclidean plane equals π. The nineteenth century version applies to a compact surface S with smooth boundary ∂S and Euler characteristic χ(S): � S � KdA + ∂S κg(s)ds = 2πχ(S) where K is the Gauss curvature of S and κg is the geodesic curvature of the boundary. In this paper we survey generalizations of the Gauss-Bonnet formula to higher- dimensional manifolds, with or without boundary. Section 2 describes the work of Hopf, the contribution of Allendoerfer and Fenchel, as well as the intrinsic proof by Chern. In section 3 we present tools from Morse theory and the theory of isometric immersions, which are used in section 4 to give an extrinsic proof of the Gauss-Bonnet formula for closed manifolds embedded in Euclidean space. Section 5 generalizes these ideas to manifolds with boundary, manifolds embedded in Euclidean spheres and non- compact manifolds. While many of the results have already appeared in [10] and [5], this is to our knowledge the first study of the Gauss-Bonnet formula for submanifolds of the sphere. 1
Page 1: MORSE THEORY AND THE GAUSS-BONNET F
Page 5 and 6: space, we can define principal curv
Page 7 and 8: of curvature forms Ω = (Ωij) is
Page 9 and 10: 3.1 Morse theory review Let f : M n
Page 11 and 12: point is degenerate if the Hessian
Page 13 and 14: Since we are working with an embedd
Page 15 and 16: Lemma 4.4. If u ∈ ν 1 M is based
Page 17 and 18: Apply the previous corollary to the
Page 19 and 20: ut falls outside the scope of this
Page 21 and 22: and the map F : ν π M → S N , F
Page 23 and 24: dp of index k. By theorem 3.2, �
Page 25: References [1] C.B. Allendoerfer, T

MORSE THEORY AND THE GAUSS-BONNET FORMULA Alina ...

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