Shape spaces of nonlinear flags

Ioana Ciuclea; Alice Barbara Tumpach; Cornelia Vizman

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arXiv · 2303.15184

Shape spaces of nonlinear flags

Ioana Ciuclea·Alice Barbara Tumpach·Cornelia Vizman

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Abstract

The gauge invariant elastic metric on the shape space of surfaces involves the mean curvature and the normal deformation, i.e. the sum and the difference of the principal curvatures $\kappa_1,\kappa_2$. The proposed gauge invariant elastic metrics on the space of surfaces decorated with curves involve, in addition, the geodesic and normal curvatures $\kappa_g,\kappa_n$ of the curve on the surface, as well as the geodesic torsion $\tau_g$. More precisely, we show that, with the help of the Euclidean metric, the tangent space at $(C,\Sigma)$ can be identified with $C^\infty(C)\times C^\infty(\Sigma)$ and the gauge invariant elastic metrics form a 6-parameter family that we give explicitly.

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Ioana Ciuclea, Alice Barbara Tumpach, Cornelia Vizman. 2023-03-27. Shape spaces of nonlinear flags. https://arxiv.org/abs/2303.15184

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