Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A fluid dynamic formulation of the isometric embedding problem in differential geometry

Authors: Gui-Qiang Chen, Marshall Slemrod and Dehua Wang
Journal: Quart. Appl. Math. 68 (2010), 73-80
MSC (2000): Primary 35M10, 76H05, 76N10, 76L05, 53C42
DOI: https://doi.org/10.1090/S0033-569X-09-01142-1
Published electronically: October 20, 2009
MathSciNet review: 2598881
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Abstract | References | Similar Articles | Additional Information

Abstract: The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.

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Additional Information

Gui-Qiang Chen
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: gqchen@math.northwestern.edu

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: slemrod@math.wisc.edu

Dehua Wang
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: dwang@math.pitt.edu

DOI: https://doi.org/10.1090/S0033-569X-09-01142-1
Keywords: Isometric embedding, two-dimensional Riemannian manifold, differential geometry, transonic flow, gas dynamics, viscosity method, compensated compactness.
Received by editor(s): August 6, 2008
Published electronically: October 20, 2009
Dedicated: Dedicated to Walter Strauss on the occasion of his 70th birthday
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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