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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(s):
Wilfried
H.
Paus
Journal:
Trans. Amer. Math. Soc.
350
(1998),
3943-3966.
MSC (1991):
Primary 58G03;
Secondary 58A15, 53C21
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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
Email:
wilfried.paus@zentrale.deuba.com
DOI:
10.1090/S0002-9947-98-02016-9
PII:
S 0002-9947(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
Copyright of article:
Copyright
1998,
American Mathematical Society
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