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

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- by Vladimir I. Oliker
- Proc. Amer. Math. Soc.
**138**(2010), 4075-4080 - DOI: https://doi.org/10.1090/S0002-9939-2010-10439-2
- Published electronically: June 29, 2010
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## 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 ${\mathbb R}^{n+1}$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.## References

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## Bibliographic Information

**Vladimir I. Oliker**- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- Email: oliker@mathcs.emory.edu
- 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
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2679628