A nonlinear Fubini-Lebesgue theorem identifies L^p curves in nonlinear Lebesgue spaces with maps into spaces of L^p curves, yielding pointwise length, curvature bounds, and speed for absolutely continuous curves.
Manifolds of absolutely continuous curves and the square root velocity framework
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abstract
A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian manifolds. As an application we consider extensions of the square root velocity transform (SRVT) framework for shape analysis. Computations in this framework frequently lead to curves which leave the shape space (of smooth curves), and are only contained in a completion. In the vector valued case, this extends the SRVT to a space of absolutely continuous curves. We investigate the situation for shape spaces of manifold valued (absolutely continuous) curves.
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math.DG 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Nonlinear Lebesgue spaces: Curves and geometry
A nonlinear Fubini-Lebesgue theorem identifies L^p curves in nonlinear Lebesgue spaces with maps into spaces of L^p curves, yielding pointwise length, curvature bounds, and speed for absolutely continuous curves.