On tight immersions of maximal codimension

Abstract

Let f : M--+ be an immersion of a compact differentiable manifold of dimension n into a Euclidean space of dimension m. The immersion f is called tight if there exists no immersion of with smaller total Lipschitz-Killing curvature [3] in any Euclidean space. The immersion f is called substantial if f(M) is not contained in any hyperplane of E =. Kuiper [4] has shown that if f is both tight and substantial, then m<=N= 89 In case m=N and n = 2 (so that N=5) , he has shown that i f f is tight and substantial then M z must be diffeomorphic to the real projective plane and f must be an embedding onto a real algebraic variety, in fact onto a Veronese surface. In this paper we prove the corresponding result in higher dimensions. Our hypothesis is, in fact, weaker. The immersion f is said to have the two-piece property if every hyperplane divides it into at most two pieces, or more exactly, if for every hyperplane HOE, f l ( H 0 and f l ( H 2 ) are both connected sets, where H~ and H 2 a r e the two open half-spaces which make up the complement of H in E'. A tight immersion has the two-piece property, but not necessarily conversely (cf. [-6,9]). However, for the case of curves and surfaces the two properties are equivalent. Let A be a real vector space of dimension n + 1 and consider the map v~-, v| from A to A| Take a metric in A and restrict the map to the unit sphere centered at the origin. Since ( -v ) |174 this map takes each pair of antipodal points to the same point. Hence it induces a map of the real projective n-space into A | A. As we shall see, this last map is an embedding, and the image Vlies substantially in an affine subspace of dimension N = 89 n (n + 3). W e call any submanifold projectively equivalent to V and lying in an affine or projective space a Veronese n-rnan~)ld. Any Veronese manifold is tightly embedded [4]. We can now state our main result.