Abstract
This work derives the basic balance laws of Codazzi, Ricci, and Gauss for the isometric embedding of an n-dimensional Riemannian manifold into the \(m=\frac{n}{2}\left( n+1\right) \)-dimensional Euclidean space. It is shown how the balance laws can be expressed in quasi-linear symmetric form and how weak solutions for the linearized problem can be established from the Lax-Milgram theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Deane Yang pointed out this equality to me and called it a “gauge condition ”. An analogy with continuum mechanics might be setting the pressure equal to zero on the surface of a water wave.
References
Blum R (1946) Ueber die Bedingungsgleichungen einer Riemann’schen Mannigfaltiggkeit, die in einer Euklidischen Mannigfaltigkeit enigebetter ist. (in German). Bull Math Soc Roum 47:144–201
Blum R (1947) Sur les tenseurs dérivés de Gauss et Codazzi. C R Acad Sci Paris 244:708–709
Blum R (1955) Subspaces of Riemannian spaces. Can J Math 7:445–452
Bryant RL, Griffiths PA, Yang D (1983) Characteristics and existence of isometric embeddings. Duke Math J 50:893–994
Friedrichs KO (1956) Symmetric positive linear differential equations. Commun Pure Appl Math 11:333–418
Goenner HF (1977) On the inderdependency of the Gauss-Codazzi-Ricci equations of local isometric embedding. Gen Relativ Gravit 8:139–145
Goodman J, Yang D (1988) Local solvability of nonlinear partial differential equations of real principal type. Unpublished www.deaneyang.com/paper,goodman-yang.pdf
Han Q, Khuri M (2012) The linearized system for isometric embeddings and its characteristic variety. Adv Math 23:263–293
Han Q, Hong J-X (2006) Isometric embedding of Riemannian manifolds in Euclidean spaces. American Mathematical Society, Providence
Nakamura G, Maeda Y (1986) Local isometric embedding problem of Riemannian \(3\)-manifolds into \({\rm {I}}\!{\rm {R}}^{6}\). Proc Jpn Acad Ser A Math Sci 62:257–259
Nakamura G, Maeda Y (1989) Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces. Trans Am Math Soc 313:1–51
Poole TE (2010) The local isometric embedding problem for \(3\)-dimensional Riemannian manifolds with cleanly vanishing curvature. Commun Partial Differ Equ 35:1802–1826
Nash JF Jr (1956) The embedding problem for Riemannian manifolds. Ann Math 63:20–63
Yau S-T (2006) Perspectives on geometric analysis [arXiv:math/0602363, vol 2, 16 Feb. 2006]: also Proceedings of International Conference on Complex Geometry and Related Fields, AMS/IP. Stud Adv Math., vol 39, pp 289–378. American Mathematical Society, Providence (2007)
Yosida K (1965) Functional analysis. Springer, Berlin
Acknowledgments
I would like to thank my co-investigators on the research project on Higher Dimensional Isometric Embedding: G.-Q. Chen (Oxford), J.Clelland (Colorado), D.Wang (Pittsburgh), and D.Yang (Poly-NYU) for their many comments and suggestions over several years. Our project commenced in Palo Alto, California, at the American Institute for Mathematics (AIM). Financial support from Estelle Basor and Brian Conrey via the AIM SQuaRE’s program has been especially helpful. In addition, my research has been supported by the Simons Foundation Collaborative Research Grant 252531 and the Korean Mathematics Research Station at KAIST (Daejeon, S.Korea). In fact, these notes were the basis for a lecture series at KAIST given at the kind invitation of Professor Y.-J. Kim. Additional thanks are due to Keble College (Oxford) where I was a visitor in April 2013 and had further opportunity to complete these notes. Very special thanks are extended to Irene Spencer and Mary McAuley of the Department of Mathematics and Statistics, University of Strathclyde (Glasgow) for the wonderful success in transforming my moderately intelligible hand written draft into the present “tex” version. Finally, I would like to express my gratitude to the organizers of the ICMS (Edinburgh) Workshop “Differential Geometry and Continuum Mechanics” . These are Jack Carr, G.-Q. Chen, M.Grinfeld, R.J. Knops, and J. Reese. Indeed, it was Michael Grinfeld who kindly arranged for Irene and Mary to type and organize these notes.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Slemrod, M. (2015). Lectures on the Isometric Embedding Problem \((M^{n},g)\rightarrow \mathrm {I\!R\!}^{m},\,m=\frac{n}{2}(n+1)\) . In: Chen, GQ., Grinfeld, M., Knops, R. (eds) Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics & Statistics, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-319-18573-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-18573-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18572-9
Online ISBN: 978-3-319-18573-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)