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index(p) = 2, index(q) = index(r) = index(a) = 1, index(s) = 0, and we obtain χ(M) = (−1) 0 · 1 + (−1) 1 · 3 + (−1) 2 · 1 = −1 as expected. Example. Consider the annulus M ⊂ R 2 and the height function h representing the y-coordinate. There are four critical points, all on the boundary, but only two of them are used for computing χ(M) = (−1) 0 · 1 + (−1) 2 · 1 = 0. 3.2 Isometric immersions and the Gauss map Let M n be a submanifold of R N . The unit normal bundle of M is a bundle with fiber S N−n−1 and total space of dimension N − 1 defined as: ν 1 M = {(x, u) : x ∈ M, u ∈ TxR N , u ⊥ TxM and �u� = 1} For any (x, u) ∈ ν 1 M, the tangent space T(x,u)ν 1 M splits into a horizontal subspace and a vertical subspace: T(x,u)ν 1 M = Hor T(x,u)ν 1 M ⊕Vert T(x,u)ν 1 M. In the canonical metric on ν 1 M the horizontal and vertical subspaces are orthogonal to each other. This splitting gives a canonical volume form dV ν 1 M on ν 1 M which locally coincides with dVM ∧ dV S N−n−1. 10
Since we are working with an embedded manifold, we will also need to consider the ”shape” of the embedding, given by the second fundamental form. If f : M → N is an isometric embedding, we define the second fundamental form (or shape operator in [6]) to be a map that associates to every normal vector u ⊥ TxM the unique self-adjoint operator Au : TxM → TxM satisfying 〈Au(v), w〉 = 〈∇V W, u〉. Here V and W are extensions of v and w, respectively, to vector fields tangent to M in a small neighborhood of x, and ∇ is the covariant derivative in the ambient manifold N. One can show that this definition does not depend on the extensions chosen and that Au(v) = −(∇vU) T where U is a vector field normal to M extending u on a small neighborhood and T denotes the tangential component of the derivative. Again, this equality does not depend on the extension U. As a self-adjoint operator, Au has real eigenvalues κ1, ..., κn called the principal curvatures of the embedded submanifold M in the direction of u. The determinant det Au = κ1 · ... · κn is a generalization of the Gaussian curvature of a surface. In the case of a hypersurface, det Au is called the Gauss-Kronecker curvature. Recall that M is a submanifold of R N . For some x ∈ M we can interpret TxM as an n-subspace of R N , and view ν 1 xM as the set of all unit vectors in R N orthogonal to TxM. Then we can define the Gauss map G : ν 1 M → S N−1 ⊂ R N by G(x, u) = u. It is known (see [3] or [10]) that G ∗ (dVS)G(x,u) = |det Au|(dV ν 1 M)u for all u ∈ ν 1 M. In particular, u ∈ ν 1 M is a regular point for G if and only if Au is invertible. 11
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: point is degenerate if the Hessian
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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