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Fenchel type theorem for submanifolds of \(S^ n\). (English) Zbl 0880.53047

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. [E. Teufel, Manuscr. Math. 75, 43-48 (1992; Zbl 0754.53009)].

MSC:

53C40 Global submanifolds
53C65 Integral geometry

Citations:

Zbl 0754.53009
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