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Transactions of the American Mathematical Society

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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
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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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

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