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some scaling. Therefore, any compact manifold can be isometrically embedded in an Euclidean sphere if we allow scaling. 5.3 Noncompact complete manifolds Another possible generalization is to drop the assumption of compactness. Consider a complete noncompact surface S. We should require that χ(S) is finite, which implies that S must have finite topological type: it must have finite genus and only finitely many ends. Also, we should require that S has an absolutely integrable Gaussian � � curvature: |K|dA < ∞ in order to guarantee that the integral KdA converges S and is finite. In this context we have the following: � Theorem 5.4. Cohn-Vossen. 2πχ(S) − for any complete, noncompact surface. S � KdA ≥ 0. In particular, S S 22 KdA ≤ 2π The Gauss-Bonnet formula might still hold in some cases, or one can study the � curvature defect 2πχ(S) − KdA. For a recent study of this area and higher dimen- sional generalizations, see [5]. S
References [1] C.B. Allendoerfer, The Euler number of a Riemann manifold, Amer. J. Math. 62 (1940), 243– 248. [2] S.S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747–752. MR 0011027 (6,106a) [3] S.S. Chern and R.K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306–318. [4] , On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5–12. [5] F. Dillen and W. Kühnel, Total curvature of complete submanifolds of Euclidean space, Tohoku Math. J. (2) 57 (2005), no. 2, 171–200. MR 2137465 (2006e:53095) [6] M. P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992. [7] W. Fenchel, On total curvatures of Riemannian manifolds: I, J. London Math. Soc. 15 (1940), 15–22. [8] A. Gray, Tubes, 2nd ed., Progress in Mathematics, vol. 221, Birkhäuser Verlag, Basel, 2004. [9] J. Milnor, Morse theory, Princeton University Press, Princeton, N.J., 1963. [10] D.V. Tausk, F. Mercuri, and P. Piccione, Notes on Morse theory, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2001. [11] H. Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939), no. 2, 461–472. 23
Page 1 and 2: MORSE THEORY AND THE GAUSS-BONNET F
Page 3 and 4: 1 Introduction The simplest version
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: dp of index k. By theorem 3.2, �

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