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Mesures de courbure des variétés lisses et des polyèdres (d’après Cheeger, Miller et Schrader). (Measure of curvature of smooth manifolds and of polyhedra (according to Cheeger, Miller and Schrader)). (French) Zbl 0613.53031
Sémin. Bourbaki, 38ème année, Vol. 1985/86, Exp. No. 664, Astérisque 145/146, 241-256 (1987).
[For the entire collection see Zbl 0601.00002.]
This exposé is a condensed and very clear presentation of the main result on curvatures of piecewise linear spaces corresponding to certain ”Lipschitz-Killing curvatures” of Riemannian spaces \((= trace\) of powers of the curvature operator) due to J. Cheeger, W. Müller and R. Schrader [Commun. Math. Phys. 92, 405-454 (1984; Zbl 0559.53028); see also Indiana Univ. Math. J. 35, 737-754 (1986)]. It says that for ”sufficiently fat” polyhedral approximations of Riemannian spaces these PL curvatures converge (in the measure sense) to the corresponding curvature. This holds for Lipschitz-Killing curvatures of even order including the Gauss-Bonnet integrand of even dimensional Riemannian manifolds. This leads to a new proof of the Gauss-Bonnet-Chern formula. In a final chapter the author indicates shortly the relation with the PL version of the Laplace operator on p-forms and the corresponding formula for \(\eta\)-invariant and signature due to J. Cheeger [J. Differ. Geom. 18, 575-657 (1983; Zbl 0529.58034)].
Reviewer: W.Kühnel
MSC:
53C65 Integral geometry
57Q15 Triangulating manifolds
52A22 Random convex sets and integral geometry (aspects of convex geometry)
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