-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

Full-text PDF

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 -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.

**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****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**, 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**, 10.1137/S0036139995287685

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
58E50,
53C22

Retrieve articles in all journals with MSC (2000): 58E50, 53C22

Additional Information

**Jayant Shah**

Affiliation:
Mathematics Department, Northeastern University, Boston, Massachusetts

Email:
shah@neu.edu

DOI:
https://doi.org/10.1090/S0033-569X-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.