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
Full-text PDF Free Access
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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.
References
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References
- G.-Q. Chen, C. Dafermos, M. Slemrod, and D. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys. 271 (2007), 635-647. MR 2291790 (2008e:35149)
- G.-Q. Chen, M. Slemrod, and D. Wang, Vanishing viscosity method for transonic flow, Arch. Rational Mech. Anal. 189 (2008), 159-188. MR 2403603
- G.-Q. Chen, M. Slemrod, and D. Wang, Isometric immersions and compensated compactness, submitted.
- M. P. do Carmo, Riemannian Geometry, Transl. by F. Flaherty, Birkhäuser: Boston, MA, 1992. MR 1138207 (92i:53001)
- G.-C. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs, 95, American Mathematical Society, Providence, RI, 1991.
- R. Finn and D. Gilbarg, Uniqueness and the force formulas for plane subsonic flows, Trans. Amer. Math. Soc. 88 (1958), 375–379.
- F. I. Frankl, M. V. Keldysh, Die äussere Neumann’sche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendung auf die Theorie der Flügel im kompressiblem Gas (Russian, German summary), Izvestiya Akademii Nauk SSR, Series 7 (1934), no. 4, 561-607.
- M. Gromov, Partial Differential Relations, Springer-Verlag: Berlin, 1986. MR 864505 (90a:58201)
- Q. Han, and J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS: Providence, RI, 2006. MR 2261749 (2008e:53055)
- J.-X. Hong, Realization in $\mathbb {R}^3$ of complete Riemannian manifolds with negative curvature, Comm. Anal. Geom. 1 (1993), no. 3-4, 487–514. MR 1266477 (95d:53003)
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Additional Information
Gui-Qiang Chen
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208
MR Author ID:
249262
ORCID:
0000-0001-5146-3839
Email:
gqchen@math.northwestern.edu
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
MR Author ID:
163635
Email:
slemrod@math.wisc.edu
Dehua Wang
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email:
dwang@math.pitt.edu
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.