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 -type Riemannian metrics on the space of planar curves
Author(s):
Jayant
Shah
Journal:
Quart. Appl. Math.
66
(2008),
123-137.
MSC (2000):
Primary 58E50;
Secondary 53C22
Posted:
December 6, 2007
MathSciNet review:
2396654
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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.
References:
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- P. Michor and D. Mumford, ``Riemannian geometries on spaces of plane curves'', arXiv:math.DG/0312384, v2, Sep. 22, 2004. MR 2201275 (2007a:58007)
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Additional Information:
Jayant
Shah
Affiliation:
Mathematics Department, Northeastern University, Boston, Massachusetts
Email:
shah@neu.edu
PII:
S0033-569X-07-01084-4
Keywords:
Moduli of planar curves,
Differential geometry
Received by editor(s):
February 27, 2007
Posted:
December 6, 2007
Additional Notes:
This work was supported by NIH Grant I-R01-NS34189-08.
Copyright of article:
Copyright
2007,
Brown University
The copyright for this article reverts to public domain after 28 years from publication.
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