zbMATH — the first resource for mathematics

\(H^{o}\)-type Riemannian metrics on the space of planar curves. (English) Zbl 1144.58005
Summary: 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.

58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53C22 Geodesics in global differential geometry
Full Text: DOI Link arXiv
[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. · Zbl 0613.53001
[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. · Zbl 1101.58005 · doi:10.4171/JEMS/37 · doi.org
[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. · Zbl 1116.58007
[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. · Zbl 0907.68158 · doi:10.1137/S0036139995287685 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.