-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 -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.

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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

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.