Curvaturas absolutas totales de variedades contenidas en un espacio euclidiano

Abstract

Let Xn be a compact (without boundary) differentiable manifold of dimensión n and class C" contained in the euclidean space En*N. We define tlie foliowing total absolute curvatures Kr(Xn), ( l ^ r ¿ n + N — 1): a) Case 1 ¿ r ¿ n . Let Tn (p) be tlie tangent space of X" at tlie point p . Let Ln + N-r (0) be a (n -I- N — r) — dimensional linear subspace througti the fixed point Oand let dLn + N-r (0) be tlie density for sets of Ln + N-r(0) (volume element of the Grassmann manifold Gn + N-r,r). Let Tr denote the set of all linear subspaces Lr of E» + N which are contained in some Tn (p), pass through p and are orthogonal to Ln + N-r(0). Then we define Kr (Xn) by the formulae (1 . 1), (1 .2). b) Case n ^ r ^ n - f - N — 1. With the same notations above, denoting now Fr the set of all linear subspaces Lr which contain some Tn (p) and are orthogonal to Ln+N-r (0), the total absolute curvature Kr(Xn) is defined by the same formulae (1.1) and (t . 2). We prove the foliowing properties of these curvatures: 1. In case 1 ¿ r ¿ n we have Kr (X" ) ^í O if and only if n ^ r N; 2. For r ^ n the only absolute curvature which is 7^ O is Kn + N-i (X» ) and it coincides with the curvature of Chern-Lashof; 5. The case N = 1 generalizes to compact manifolds the well known mean curvatures of convex hypersurfaces; 4. The case n — r N (formula (2.9)) Is particularlly interesting; we consider in detall the case n — 2, N = 2, r—1. In n.° 4 we state some inequalities among the absolute curvatures Kr (X" )