Eberhard Teufel

A generalization of the isoperimetric inequality in the hyperbolic plane

where the curve is given by a smooth immersion f of a circle into E and w I (p) denotes the winding number of f w.r.t, p ~ E. Equality holds if and only if the curve is a circle traversed in the same direction a number of times. (In particular Banchoff and Pohl derived an inequality of this type which applies to compact immersed submanifolds in euclidean spaces of arbitrary dimensions.) (Cf. also [3] for a polygonal version of (2).) In the spherical case (1) can be written more symmetrically, namely

Differential Topology and the Computation of Total Absolute Curvature

The total absolute (Lipschitz-Killing-)curvature of an immersion of a manifold in Euclidean space is equal to the mean value of the number of critical points of certain level functions which are induced by bundles of parallel hyperplanes (see Chern-Lashof and Kuiper). We generalize this for immersions in spaces of constant curvature + 1. A level function is induced by an oriented hyperplane bundle either through a common plane of codimension two or orthogonal to a geodesic. In the hyperbolic case we have to restrict the level function to the interior of the normal tube of constant radius around that plane or geodesic.

Eine differentialtopologische Berechnung der Totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie

The aim of this article is to compute the total (absolute) curvature, i.e. the mean value of the (absolute) Lipschitz-Killing-curvature, of an immersion f: M→Sn of a compact manifold into the unit sphere in a differential topological manner. Through a generalization of KUIPERs treatment of immersions in Euclidean spaces it can be computed as the mean value of the number of critical points—weighted by (−1)k (k=Index) resp.not weighted—of certain functions. These functions are the pullback via f of “level-functions”, which are defined almost everywhere on Sn. Such a “level-function” is constructed by taking any oriented great circle as a “leveling-scale” and the orthogonal great (n−1)-spheres as “level-surfaces”.

On the total absolute curvature of closed curves in spheres

The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchels inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnors inequality, 1949, [2], [7], [5], [4]).For the total absolute curvature of closed curves in spheres no such lower bounds exist because there are closed geodesies. Here we derive similar bounds which depend on the length of the curve resp.the area of surfaces of disk-type bounded by the curve.In order to prove these inequalities we start from the computation of the total absolute curvature as mean value of the number of critical points of certain level functions ([11],[12]); we use some topological considerations and Poincarés integralgeometric formula for the computation of length resp. area.

Anwendungen der differentialtopologischen Berechnung der totalen Krümmung und totalen Absolutkrümmung in der sphärischen Differentialgeometrie.

In a previous article [16] we have shown how the total absolute (Lipschitz-Killing) curvature of the immersion f:M→Sn of a compact manifold into a sphere can be computed in a differential topological manner as the mean-value of the number of critical points of certain level-functions. (And similar the total curvature.) (comp. [12], prob. 15) Now we consider the gradient vector field of the level-functions and achieve a relation between the total curvature and the Euler characteristic, of the manifold, which can be sharpened in some cases to inequalities. Moreover it leads to the formula of Allendoerfer-Weil for compact n-dim. submanifolds of the sphere Sn.