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Transactions of the American Mathematical Society

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

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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
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by Wilfried H. Paus PDF
Trans. Amer. Math. Soc. 350 (1998), 3943-3966 Request permission

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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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
  • 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
  • © Copyright 1998 American Mathematical Society
  • 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