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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
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References:
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