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Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

$ H^{o}$-type Riemannian metrics on the space of planar curves


Author: Jayant Shah
Journal: Quart. Appl. Math. 66 (2008), 123-137
MSC (2000): Primary 58E50; Secondary 53C22
Published electronically: December 6, 2007
MathSciNet review: 2396654
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Abstract | References | Similar Articles | Additional Information

Abstract: Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for sectional curvature for conformally equivalent metrics. We show if the conformal factor depends only on the length of the curve, then the metric behaves like an $ L^1$-metric, the sectional curvature is not bounded from above, and minimal geodesics may not exist. If the conformal factor is superlinear in curvature, then the sectional curvature is bounded from above.


References [Enhancements On Off] (What's this?)

  • 1. Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684 (88f:53087)
  • 2. Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 1, 1–48. MR 2201275 (2007a:58007), http://dx.doi.org/10.4171/JEMS/37
  • 3. P. Michor and D. Mumford, ``An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach'', Tech. Report, ESI Preprint #1798, 2005.
  • 4. E. Klassen, A. Srivastava, W. Mio, and S. H. Joshi, ``Analysis of planar shapes using geodesic paths on shape spaces'', IEEE Trans. PAMI, 26(3), pp. 372-383, 2003.
  • 5. W. Mio and A. Srivastava, ``Elastic-string models for representation and analysis of planar shapes'', CVPR(2), 2004, pp. 10-15.
  • 6. W. Mio, A. Srivastava, and S. H. Joshi, ``On shape of plane elastic curves'', International Journal of Computer Vision, 73(3), pp. 307-324.
  • 7. A. Yezzi and A. Mennucci, ``Conformal Riemannian metrics in space of curves'', EUSIPCO04, MIA, 2004.
  • 8. A. Yezzi and A. Mennucci, ``Metrics in the space of curves'', arXiv:math.DG/0412454, v2, May 25, 2005.
  • 9. Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math. 58 (1998), no. 2, 565–586 (electronic). MR 1617630 (99c:68221), http://dx.doi.org/10.1137/S0036139995287685

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Additional Information

Jayant Shah
Affiliation: Mathematics Department, Northeastern University, Boston, Massachusetts
Email: shah@neu.edu

DOI: http://dx.doi.org/10.1090/S0033-569X-07-01084-4
PII: S 0033-569X(07)01084-4
Keywords: Moduli of planar curves, Differential geometry
Received by editor(s): February 27, 2007
Published electronically: December 6, 2007
Additional Notes: This work was supported by NIH Grant I-R01-NS34189-08.
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.



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