Fenchel type theorem for submanifolds of \(S^ n\)
The main result is: Let \(M\) be a compact hypersurface in the \(n\)-dimensional sphere, then \[ \sum^{n-1}_{i=0} c_iL_i (M)\geq \beta (M), \] where \(\beta (M)=\) sum of the Betti numbers of \(M\), \(L_i =i\)-length functional, defined integral-geometrically and localized through absolute curvature terms of \(M\), \(c_i=\) dimension constants. Further results of this nature deal with knotted curves and surfaces in spheres. Remark: Concerning the optimal Fenchel type theorem for curves in spheres cf. [\textit{E. Teufel}, Manuscr. Math. 75, 43-48 (1992; Zbl 0754.53009)].
scientific article; zbMATH DE number 1008571