Fractional Sobolev metrics on spaces of immersed curves

  • Martin Bauer
  • Martins BruverisEmail author
  • Boris Kolev


Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) and on its Sobolev completions \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\). We prove local well-posedness of the geodesic equations both on the Banach manifold \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) and on the Fréchet-manifold \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) provided the order of the metric is greater or equal to one. In addition we show that the \(H^s\)-metric induces a strong Riemannian metric on the Banach manifold \({\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) of the same order s, provided \(s>\frac{3}{2}\). These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.

Mathematics Subject Classification

58D05 35Q35 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty for MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsBrunel University LondonUxbridgeUK
  3. 3.UMR 7373, I2M, Centrale Marseille, CNRSAix Marseille UniversitéMarseilleFrance

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