Algebraic Topology, Barcelona 1986 by J. Aguade, R. Kane

By J. Aguade, R. Kane

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This does not contradict Filling Lemma where we assume n > 3, IDEA OF THE PROOF. Given an immersed locally convex hyper surf ace in V, we may try the inward locally equidistant deformation which applies simultaneously to all small embedded neighbourhoods of e(W) c V, see Fig. 18 below. The equidistant deformation sketched in Fig. 18 develops a (cuspidal) singularity at a certain moment &, and cannot be continued beyond e0. However, an elementary argument as in § L'a shows that no such singuarities appear for deformations of locally convex (pieces of) hypersurfaces in 1R" for n > 3, This conclusion extends to all Riemannian manifolds V with local geodesic coordinates systems at the points x£e(S) where one is afraid of singularities.

REMARK. The Gauss-Bonnet theorem generalizes to all dimensions by where £i = Q (v) is expressible at each v as a certain polynomial in the components of the curvature tensor. One knows that for dim F = 4 the sign conditions i £ > 0 and K < 0 both imply -Q(u)> 0It follows that if the curvature of V does not change sign then 0. This is not very interesting for K > 0, where the universal covering V of V is compact and where 3i1{V)^r=0=^b1_=ba=0. So the remaining Betti numbers contributing to x(Tr) = x(^) a r e even: &„, 63 and &4.

Here V—{V,g) is a complete Rieraannian manifold with K{V) < 0. One can easily derive from the tube formula (see (**) in § 2) that the condition K < 0 is equivalent to the preservation of convexity under outward equidistant deformations fF£>« of convex hyper surf aces W in V, for small e, see Fig. 19. W Pig. 19. This is quite similar to the case K > 0. e. an immersion) for all e > G and d,(W)*->W is a locally convex immersed hypersurface. The only problem comes from possible self-intersections of this hypersurface.

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