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Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields


Author: Wilfried H. Paus
Journal: Trans. Amer. Math. Soc. 350 (1998), 3943-3966
MSC (1991): Primary 58G03; Secondary 58A15, 53C21
DOI: https://doi.org/10.1090/S0002-9947-98-02016-9
MathSciNet review: 1443201
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Abstract: In this paper, we investigate under what circumstances the Laplace–Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space. We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat. These results are achieved by formulating the problem as an exterior differential system and applying the Cartan–Kähler theorem to it.


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  • R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687
  • Pierre H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), no. 3, 249–276. MR 455055, DOI https://doi.org/10.1007/BF02028444
  • R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148
  • Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
  • Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
  • Lawrence Conlon, Differentiable manifolds: a first course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1209437
  • M.J. Cornwall, Brownian Motion and Heat Kernels on Lie Groups, Ph.D. thesis, University of New South Wales, Sydney, 1994.
  • Georges de Rham, Differentiable manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 266, Springer-Verlag, Berlin, 1984. Forms, currents, harmonic forms; Translated from the French by F. R. Smith; With an introduction by S. S. Chern. MR 760450
  • F. G. Friedlander, The wave equation on a curved space-time, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Cambridge Monographs on Mathematical Physics, No. 2. MR 0460898
  • S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR 0424186
  • Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI https://doi.org/10.1007/BF02392081
  • Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
  • Serge Lang, Differential and Riemannian manifolds, 3rd ed., Graduate Texts in Mathematics, vol. 160, Springer-Verlag, New York, 1995. MR 1335233
  • Daniel Martin, Manifold theory, Ellis Horwood Series in Mathematics and its Applications: Statistics, Operational Research and Computational Mathematics, Ellis Horwood, New York, 1991. An introduction for mathematical physicists. MR 1108624
  • Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276
  • Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
  • W.H. Paus, Sum of squares manifolds: The expressibility of the Laplace–Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields, Ph.D. thesis, University of New South Wales, Sydney, 1996.
  • Joseph Fels Ritt, Differential algebra, Dover Publications, Inc., New York, 1966. MR 0201431
  • David Ruelle, États d’équilibre des systèmes infinis en mécanique statistique, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 15–19. MR 0436834
  • Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983), no. 1, 48–79. MR 705991, DOI https://doi.org/10.1016/0022-1236%2883%2990090-3
  • J. C. Taylor, The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of noncompact type, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 303–332. MR 954647, DOI https://doi.org/10.1090/conm/073/954647
  • N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
  • Joseph A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Inc., Boston, Mass., 1974. MR 0343214
  • Hung Hsi Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i–xii and 289–538. MR 1079031

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Additional Information

Wilfried H. Paus
Affiliation: School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
Address at time of publication: Deutsche Bank AG, Credit Risk Management, 60262 Frankfurt am Main, Germany
Email: wilfried.paus@zentrale.deuba.com

Keywords: Differential geometry, pseudo-Riemannian manifolds, the Laplace–Beltrami operator, exterior differential systems, Cartan–Kähler
Received by editor(s): December 30, 1996
Additional Notes: This work was made possible through funding from the Australian Department of Employment, Education and Training (OPRS), the Deutscher Akademischer Austauschdienst of Germany, the Australian Research Council Grant “Differential and Integral Operators”, and the Deutsche Forschungsgemeinschaft.
Dedicated: To my aunt Ingrid S. Keller
Article copyright: © Copyright 1998 American Mathematical Society