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

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gave an intrinsic proof of the formula. His paper [2] was so well received that Allendoerfer and Fenchel’s formula is now known as the Gauss-Bonnet-Chern formula. 2.2 Chern’s intrinsic proof: a sketch As above, let M 2n be a compact, even-dimensional manifold and consider its unit tangent bundle UM. Let π be the projection map UM → M. Chern proved that the pullback of the 2n-degree Euler form ω = 1 (2π) n/2 Pf(Ω) to UM is an exact form: π ∗ (ω) = dΦ for some (2n − 1)-form Φ on UM. If V is a smooth vector field on M with isolated non-degenerate zeroes then the vector field X = V �V � has only isolated singularities; let U be the set of singularities of X on M. We can view X as a section X : M \ U → UM. Chern proved that the boundary of the image X(M \ U) is an (2n − 1)-dimensional cycle of UM. Then, by Stokes’ theorem: � � ω = X ∗ � (dΦ) = M M\U X(M\U) � dΦ = ∂X(M\U) With some work one can show that the last integral is exactly the sum of the indices of the vector field X. By theorem 2.1, this is equal to χ(M). 3 Tools: Morse Theory and Integral Geometry For the remainder of this paper we drop the requirement that M has even dimension. Φ 6
3.1 Morse theory review Let f : M n → R be a smooth function on a closed manifold M. We say that a point p ∈ M is a critical point for f if the induced map f∗ : TpM → Tf(p)R is zero. For such a critical point we define the Hessian of f at p as a symmetric bilinear functional f∗∗ on TpM acting on vectors as follows: if v, w ∈ TpM then consider their extensions V, W to vector fields tangent to M in a neighborhood of p and let f∗∗(v, w) = Vp(W (f)) (where Vp = v). One can check that f∗∗ is symmetric and independent of the extensions V, W . The index of a bilinear functional A on a vector space V is defined to be the maximal dimension of a subspace of V on which A is negative definite; the nullity of A is the dimension of its nullspace, the space of all vectors v such that A(v, w) = 0 for all w ∈ V . We define the index of f at p to be the index of the bilinear map f∗∗ at p. We say that p is a degenerate critical point if the nullity of f∗∗ at p is zero and a non-degenerate critical point otherwise. If (x1, ..., xn) is a local coordinate system around p, then p is critical if and only if ∂f ∂f (p) = ... = ∂x1 ∂xn (p) = 0. If v = � ai ∂ ∂xi |p and w = � bj ∂ ∂xj |p are vectors in TpM(for some constants ai, bj) then f∗∗(v, w) = � i,j aibj 7 ∂2f (p). Therefore the ∂xi∂xj matrix associated to f∗∗ with respect to the basis ∂ ∂x1 |p, ..., ∂ ∂xn |p is ( ∂2 f ∂xi∂xj (p)) and one can check that p is a nondegenerate critical point of f if and only if this matrix is nonsingular. We can also prove that the index of f∗∗ is the number of negative eigenvalues of this matrix, a number which does not depend on the coordinates chosen.
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: of curvature forms Ω = (Ωij) is
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

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