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Integral and current representation of Federer’s curvature measures. (English) Zbl 0598.53058

For arbitrary subsets of \(R^ d\) with positive reach d-1 principal curvatures (as functions on the unit normal bundle with range \((- \infty,+\infty] )\) are introduced. Federer’s curvature measures may be expressed by integrating elementary symmetric functions of the principal curvatures over the unit normal bundle with respect to the (d-1)- dimensional Hausdorff measure. This extends well-known classical results from differential geometry concerning Lipschitz-Killing curvatures of \(C_ 2\)-submanifolds. The integrals are also interpreted as values of associated rectifiable currents (normal cycles) on special differential forms.
In forthcoming papers the results are extended to normal cycles corresponding to finite unions of such sets, in particular to cell complexes.

MSC:

53C65 Integral geometry
58A25 Currents in global analysis
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