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

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Apply corollary 4.7 for the function f : ν 1 M → R, f(u) = 1; then: � AT C(M) = � = � = M � ν 1 pM SN−1 u∈G−1 (v) SN−1 k=0 � |det dGu|dudp = � 1 · dV S N−1(v) ν 1 M n� #Ck(v)dVSN−1(v) ≥ |det dGu|dV ν 1 M n� bk · Vol(S N−1 ) But b0 ≥ 1 for any manifold and bn = 1 if M is compact. Therefore, desired. k=0 16 n� bk ≥ 2 as If AT C(M) < 3 Vol(S N−1 ) then there exists a set A ⊂ S0 with positive measure such that for any v ∈ A the height function hv has less than 3 critical points on M. But if M is compact then every such hv has at least two critical points: an absolute minimum and an absolute maximum. We can conclude that all height functions hv with v ∈ A have exactly two critical points. By a theorem of Reeb’s (see [9]), if M is a compact manifold and f is a differential function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere. This completes the proof of (ii). The proof of part (iii) is more complicated. If AT C(M) = 2 Vol(S N−1 ) then hv must have exactly two critical points on M for almost all v ∈ S0. Chern and Lashof prove that if M is not contained in a linear subspace of dimension (n+1) then there is a neighborhood U ∈ S N−1 such that hv has at least three critical points for all v ∈ U. This is a contradiction and shows that M must be embedded in an (n + 1)-linear subspace. The proof that M is actually a convex hypersurface is also very interesting n=0
ut falls outside the scope of this discussion. 5 Generalizations 5.1 Manifolds with boundary Let M n be a compact submanifold of R N with boundary ∂M. Then ∂M is an embedded (n − 1)-submanifold of R N and at every point of ∂M we can define a unique outward pointing unit vector νout(x) tangent to M. The discussion in section 3.1 suggests that we should consider the bundle NM = ν 1 (M \ ∂M) � ν 1 +∂M where ν 1 (M \ ∂M) = {(x, v) : x ∈ M \ ∂M, v ∈ TxR N , v ⊥ TxM and �v� = 1} and ν 1 +∂M = {(x, v) : x ∈ ∂M, v ∈ TxR N , v ⊥ Tx∂M, �v� = 1 and 〈v, νout(x)〉 < 0} Notice that ν 1 (M \ ∂M) and ν 1 +∂M are bundles with the same total dimension N − 1. For ν 1 (M \ ∂M), the base space M \ ∂M has dimension n and the fiber has dimension N − n − 1; for ν 1 +∂M the base space ∂M has dimension n − 1, but the fiber has dimension N −n. The volume form on ν 1 (M \∂M) is locally a product dVM\∂M ∧ dV S N−n−1 and the volume form on ν 1 +∂M is locally the product dV∂M ∧dV H N−n, where H N−n denotes the round unit hemisphere of dimension N − n. We can still define a smooth Gauss map G : NM → S N−1 by G(x, u) = u as a map between manifolds of the same dimension N − 1. As in section 3, one can prove that G ∗ (dVS)G(u) = |det Au|(dVNM)u for all u ∈ NM. In this formula, Au is 17
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: Apply the previous corollary to the
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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