Taiwanese Journal of Mathematics

A LOOP GROUP FORMULATION FOR CONSTANT CURVATURE SUBMANIFOLDS OF PSEUDO-EUCLIDEAN SPACE

David Brander and Wayne Rossman

Full-text: Open access

Abstract

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\bf R_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.

Article information

Source
Taiwanese J. Math., Volume 12, Number 7 (2008), 1739-1749.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405084

Digital Object Identifier
doi:10.11650/twjm/1500405084

Mathematical Reviews number (MathSciNet)
MR2449661

Zentralblatt MATH identifier
1171.37031

Subjects
Primary: 37K25: Relations with differential geometry
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53B25: Local submanifolds [See also 53C40]

Keywords
isometric immersions space forms loop groups

Citation

Brander, David; Rossman, Wayne. A LOOP GROUP FORMULATION FOR CONSTANT CURVATURE SUBMANIFOLDS OF PSEUDO-EUCLIDEAN SPACE. Taiwanese J. Math. 12 (2008), no. 7, 1739--1749. doi:10.11650/twjm/1500405084. https://projecteuclid.org/euclid.twjm/1500405084


Export citation

References

  • J. L. Barbosa, W. Ferreira and K. Tenenblat, Submanifolds of constant sectional curvature in pseudo-Riemannian manifolds, Ann. Global Anal. Geom., 14 (1996), 381-401.
  • D. Brander, Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms, Ann. Global Anal. Geom., 32 (2007), 253-275.
  • D. Brander and J. Dorfmeister, The generalized DPW method and an application to isometric immersions of space forms, Math. Z., (2008), DOI 10.1007/s00209-008-0367-9.
  • M. Brück, X. Du, J. Park and C. L. Terng, The submanifold geometries associated to Grassmannian systems, Mem. Amer. Math. Soc., 155 (2002), no. 735, viii + 95 pp.
  • F. Burstall, U Hertrich-Jeromin, F Pedit, and U Pinkall, Curved flats and isothermic surfaces, Math. Z., 225(2) (1997), 199-209.
  • F. E. Burstall, Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, AMS/IP Stud. Adv. Math. 36 (2006), 1-82.
  • J. L. Cieśliński and Y. A. Aminov, A geometric interpretation of the spectral problem for the generalized sine-Gordon system, J. Phys. A: Math. Gen., 34 (2001), L153-L159.
  • D. Ferus and F. Pedit, Isometric immersions of space forms and soliton theory, Math. Ann., 305 (1996), 329-342.
  • A. P. Fordy and J. C. Wood (eds.), Harmonic maps and integrable systems, Aspects of mathematics, Vieweg, 1994.
  • W. Rossman, M. Umehara and K. Yamada, Irreducible constant mean curvature $1$ surfaces in hyperbolic space with positive genus, Tohoku Math. J. $(2)$, 49(4) (1997), 449-484.
  • C. L. Terng, Geometries and symmetries of soliton equations and integrable elliptic equations, arXiv preprint:0212372, (2002).
  • M. Toda, Pseudospherical surfaces via moving frames and loop groups, PhD. Thesis, University of Kansas, 2000.
  • $\_\!\_\!\_\!\_\!\_\!\_\!\_\!\_$, Initial value problems of the sine-Gordon equation and geometric solutions, Ann. Global Anal. Geom., 27(3) (2005), 257-271.