A characterization of revolution quadrics by a system of partial differential equations

Author:
Vladimir I. Oliker

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4075-4080

MSC (2010):
Primary 53A05, 53C40

DOI:
https://doi.org/10.1090/S0002-9939-2010-10439-2

Published electronically:
June 29, 2010

MathSciNet review:
2679628

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such a solution, its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in : paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.

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Additional Information

**Vladimir I. Oliker**

Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Email:
oliker@mathcs.emory.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10439-2

Keywords:
Hypersurfaces of revolution,
Obata’s theorem

Received by editor(s):
June 6, 2008

Received by editor(s) in revised form:
February 20, 2009

Published electronically:
June 29, 2010

Additional Notes:
The research of the author was partially supported by National Science Foundation grant DMS-04-05622.

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2010
American Mathematical Society