Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. R. Abraham, J.E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, Berlin, 1988. MR 89f:58001
  • 2. P.H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), 249-276. MR 56:13295
  • 3. R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths, Exterior Differential Systems, Springer, Berlin, 1991. MR 92h:58007
  • 4. I. Chavel, Riemannian Geometry: a Modern Introduction, Cambridge University Press, Cambridge, 1993. MR 95j:53001
  • 5. J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom. 17 (1982), 15-53. MR 84b:58109
  • 6. L. Conlon, Differentiable Manifolds. A First Course, Birkhäuser, Basel, 1993. MR 94d:58001
  • 7. M.J. Cornwall, Brownian Motion and Heat Kernels on Lie Groups, Ph.D. thesis, University of New South Wales, Sydney, 1994.
  • 8. G. de Rham, Differentiable Manifolds, Springer, Berlin, 1984. MR 85m:58005
  • 9. F.G. Friedlander, The Wave Equation on a Curved Space-time, Cambridge University Press, Cambridge, 1975. MR 57:889
  • 10. S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press, Cambridge, 1973. MR 54:12154
  • 11. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. MR 80k:53081
  • 12. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1968), 147-171. MR 36:5526
  • 13. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland, Amsterdam, 1989. MR 90m:60069
  • 14. S. Lang, Differential and Riemannian Manifolds, Springer, Berlin, 1995. MR 96d:53001
  • 15. D. Martin, Manifold Theory: an Introduction for Mathematical Physicists, E. Horwood, New York, 1991. MR 92g:58001
  • 16. P.J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995. MR 96i:58005
  • 17. B. O'Neill, Semi-Riemannian Geometry: with Applications to Relativity, Academic Press, New York, 1983. MR 85f:53002
  • 18. 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.
  • 19. J.F. Ritt, Differential Algebra, Dover, New York, 1966. MR 34:1315
  • 20. L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent Lie groups, Acta Math. 137 (1977), 247-320. MR 55:9771
  • 21. R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48-79. MR 84m:58138
  • 22. J.C. Taylor, The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of non-compact type, Contemp. Math. 73 (1988), 303-331. MR 89f:58139
  • 23. N. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry of Groups, Cambridge University Press, Cambridge, 1992. MR 95f:43008
  • 24. J.A. Wolf, Spaces of Constant Curvature, 3rd ed., Publish or Perish, Boston, 1974. MR 49:7958
  • 25. H.-H. Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), 289-538. MR 91h:58031

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G03, 58A15, 53C21

Retrieve articles in all journals with MSC (1991): 58G03, 58A15, 53C21


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

DOI: https://doi.org/10.1090/S0002-9947-98-02016-9
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

American Mathematical Society