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

Shah, Jayant

doi:10.1090/S0033-569X-07-01084-4

$H^{o}$-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
DOI: https://doi.org/10.1090/S0033-569X-07-01084-4
Published electronically: 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 $L^1$-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.

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