Topology is an important determinant of the behavior of a great number of condensed-matter systems, but until recently it has played a minor role in elasticity. We develop a theory for the deformations of a class of twisted non-Euclidean sheets, which have a symmetry based on the celebrated Bonnet isometry. We show that nonorientability is an obstruction to realizing the symmetry globally, and it induces a geometric phase that captures a memory analogous to a previously identified one in 2D metamaterials. However, we show that orientable ribbons can also obstruct realizing the symmetry globally. This new obstruction is mediated by how the unit normal vector winds around the centerline of the ribbon and provides conditions for constructing soft modes of deformation compatible with the topology of multiply twisted connected ribbons.

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• Elastic deformation
• Elasticity
• Geometry
• Topology