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$$H^{o}$$-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.

##### MSC:
 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 53C22 Geodesics in global differential geometry
##### Keywords:
moduli of planar curves; differential geometry
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##### References:
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