×

On the curvature of piecewise flat spaces. (English) Zbl 0559.53028

The main result is: Let \(R^ j(U)\) be the integral of the j-th Lipschitz-Killing curvature of a Riemannian manifold \(M^ n\) over a neighborhood U bounded in \(M^ n\). Let \(M^ n_{\eta}\) be a piecewise linear approximation of \(M^ n\) of fatness bounded away from zero and \(R^ j_{\eta}\) the combinatorially defined j-th Lipschitz-Killing curvature. Then under certain regularity conditions for the approximations, \(| R^ j(U)-R^ j_{\eta}(U)| \to 0\) as \(\eta\) \(\to 0\). The corresponding pointwise result is false. The proof involves constructions in linear algebra and combinatorial topology that are interesting in themselves. The paper also contains a great number of illuminating comments, and an extension of the result to manifolds with boundary.
Reviewer: H.Guggenheimer

MSC:

53C20 Global Riemannian geometry, including pinching
57Q55 Approximations in PL-topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allendoerfer, C.B., Weil, A.: The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Am. Math. Soc.53, 101-129 (1943) · Zbl 0060.38102 · doi:10.1090/S0002-9947-1943-0007627-9
[2] Banchoff, T.: Critical points and curvature for embedded polyhedra. J. Diff. Geom.1, 245-256 (1967) · Zbl 0164.22903
[3] Brin, I.A.: Gauss-Bonnet theorems for polyhedrons. Uspekhi Mat. NaukIII, 226-227 (1948) (in Russian)
[4] Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Diff. Geom.18 (1983) · Zbl 0529.58034
[5] Cheeger, J., Ebin, D.G.: Comparison theorems in Riemannian geometry. Amsterdam: North-Holland 1975 · Zbl 0309.53035
[6] Cheeger, J., Müller, W., Schrader, R.: In: Unified theories of elementary particles (Heisenberg Symposium 1981). Lecture Notes in Physics. Breitenlohner, P., Dürr, H.P. (eds.). Berlin, Heidelberg, New York: Springer 1982
[7] Cheeger, J., Müller, W., Schrader, R.: In preparation
[8] Chern, S.S.: On the kinematic formula in integral geometry. J. Math. Mech.16, 101-118 (1966) · Zbl 0142.20704
[9] Collins, P.A., Williams, R.M.: Application of Regge calculus to the axially symmetric initial-value problem in general relativity. Phys. Rev. D5, 1908-1912 (1972)
[10] Collins, P.A., Williams, R.M.: Dynamics of the Friedmann universe using Regge calculus. Phys. Rev. D7, 965-961 (1973)
[11] Collins, P.A., Williams, R.M.: Regge-calculus model for the Tolman universe. Phys. Rev. D10, 3537-3538 (1974)
[12] Donnelly, H.: Heat equation and the volume of tubes. Invent. Math.29, 239-243 (1975) · Zbl 0303.35041 · doi:10.1007/BF01389852
[13] Ebin, D.G.: In: Global analysis. Proceedings of Symposium in Pure Mathematics, Vol. XV. Providence, RI: American Mathematical Society 1970
[14] Freudenthal, H.: Simplizialzerlegung von beschränkter Flachheit. Ann. Math.43, 580-582 (1942) · Zbl 0060.40701 · doi:10.2307/1968813
[15] Fröhlich, J.: IHES preprint 1981 (unpublished)
[16] Gilkey, P.B.: The index theorem and the heat equation. Boston: Publish or Perish 1974 · Zbl 0287.58006
[17] Gilkey, P.B.: The boundary integrand in the formula for the signature and Euler characteristic of a Reimannian manifold with boundary. Adv. Math.15, 334-360 (1975) · Zbl 0306.53042 · doi:10.1016/0001-8708(75)90141-3
[18] Hartle, J.B., Sorkin, R.: Boundary terms in the action for the Regge calculus. Gen. Rel. Grav.13, 541-549 (1981) · doi:10.1007/BF00757240
[19] Hasslacher, B., Perry, M.: Spin networks are simplicial quantum gravity. Phys. Lett.103B, 21-24 (1981)
[20] Hawking, S.W.: In: General relativity, an Einstein centenary survey. Hawking, S.W., Israel, W. (eds.). Cambridge: Cambridge University Press 1979 · Zbl 0424.53001
[21] Kneser, H.: Der Simplexinhalt in der nichteuklidischen Geometrie. Deutsch. Math.1, 337-340 (1936) · JFM 62.0653.03
[22] Lewis, S.M.: Two cosmological solutions of Regge calculus. Phys. Rev. D25, 306-312 (1982) · Zbl 1267.83124
[23] Lewis, S.M.: Maryland Thesis, Preprint (1982)
[24] McCrory, C.: Stratified general position. In: Lecture Notes in Mathematics, Vol. 664. Berlin, Heidelberg, New York: Springer 1978 · Zbl 0391.57015
[25] McMullen, P.: Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Philos. Soc.78, 247-261 (1975) · Zbl 0313.52005 · doi:10.1017/S0305004100051665
[26] Minkowski, H.: In: Gesammelte Abhandlungen, Vol. II, pp. 131-229. New York: Chelsea Publ. 1967
[27] Misner, C.W., Thorn, K.S., Wheeler, J.A.: Gravitation. San Francisco: Freeman 1973
[28] Munkres, J.R.: Elementary differential topology. Princeton, NJ: Princeton University Press 1966 · Zbl 0161.20201
[29] Ponzano, G., Regge, T.: In: Spectroscopic and group theoretical methods in physics. Bloch, F., Cohen, S.G., De Shalit, A., Sambursky, S., Talmi, I. (eds.). New York: Wiley 1968
[30] Rado, T.: Length and area. New York: American Mathematical Society 1948 · Zbl 0033.17002
[31] Regge, T.: General relativity without coordinates. Nuovo Cimento19, 558-571 (1961) · doi:10.1007/BF02733251
[32] Ro?ek, M., Williams, R.M.: Quantum Regge calculus. Phys. Lett.104B, 31-37 (1981)
[33] Ro?ek, M., Williams, R.M.: The quantization of Regge calculus. Preprint (1981)
[34] Santalo, L.A.: Integral geometry and geometric probability. London: Addison-Wesley 1976
[35] Schläfli, L.: On the multiple integral \(\mathop \smallint \limits^n\) dxdy...dx, whose limits arep 1=a 1 x+b 1 y+...+h 1 z>0,p 2>0, ...,p n>0 andx 2+y 2+...+z 2<1. Q. J. Pure Appl. Math.2, 269-301 (1858)
[36] Schläfli, L.: Über die Entwickelbarkeit des Quotienten zweier bestimmter Integrale von der Form ?dxdy ... dz. J. Reine Angew. Math.67, 183-199 (1867) · ERAM 067.1745cj · doi:10.1515/crll.1867.67.183
[37] Sorkin, R.: Time-evolution problem in Regge calculus. Phys. Rev. D12, 385-396 (1975)
[38] Sorkin, R.: The electromagnetic field on a simplicial net. J. Math. Phys.16, 2432-2440 (1975) · doi:10.1063/1.522483
[39] Steiner, J.: Jber preuss. Akad. Wiss. 114-118 (1840). In: Gesammelte Werke, Vol. II, pp. 171-177, New York: Chelsea 1971
[40] Sulanke, R., Wintgen, P.: Differentialgeometrie und Faserbündel. Berlin: Deutscher Verlag der Wissenschaft 1972 · Zbl 0327.53020
[41] Warner, N.P.: The application of Regge calculus to quantum gravity and quantum field theory in a curved background. Proc. R. Soc.383, 359-377 (1982) · Zbl 0494.53054 · doi:10.1098/rspa.1982.0135
[42] Weingarten, D.: Euclidean quantum gravity on a lattice. Nucl. Phys. B210, 229-249 (1982) · doi:10.1016/0550-3213(82)90241-3
[43] Weyl, H.: On the volume of tubes. Am. J. Math.61, 461-472 (1939) · JFM 65.0796.01 · doi:10.2307/2371513
[44] Wheeler, J.A.: In: Relativity, groups, and topology. deWitt, C., deWitt, B. (eds.). New York: Gordon and Breach 1964 · Zbl 0148.46204
[45] Whitney, H.: Geometric integration theory. Princeton, NJ: Princeton University Press 1971 · Zbl 1141.28002
[46] Williams, R.M., Ellis, G.F.R.: Regge calculus and observations. I. Formalism and applications to radial motion and circular orbits. Gen. Rel. Grav.13, 361-395 (1981) · Zbl 0462.53038 · doi:10.1007/BF01025469
[47] Wintgen, P.: Normal cycle and integral curvature for polyhedra in Riemannian manifolds. Colloquia Mathematica Societatis Janos Bolya, Budapest (1978) · Zbl 0509.53037
[48] Wong, C.Y.: Application of Regge calculus to the Schwarzschild and Reissner-Nørdstrom geometries at the moment of time symmetry. J. Math. Phys.12, 70-78 (1971) · doi:10.1063/1.1665489
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.