# Condensed Matter Seminar: "Imposing Curved Shapes on Solid Sheets: Instabilities, Isometries and Asymptotic Isometries"

Imposing
a curved shape on a solid sheet, generates in it elastic stress.
This familiar motif is a consequence of Gauss’ *theorema*

*Egregium*, which posits that there
exists no isometric map between two
surfaces of different Gaussian curvatures. This coupling between geometry
(curvature) and mechanics (stress) underlies the morphological
richness observed in solid sheets, and their nontrivial response to exerted forces.

In this talk I will attempt to provide a unifying framework for this geometry-mechanics interplay, using the concept of “asymptotic isometry” that characterizes the state of a very thin sheet under very weak loads. Focusing on two problems – twisting a ribbon and poking a floating sheet – I will demonstrate how specific parameter regimes are characterized by distinct mechanisms through which the Gaussian curvature is accommodated or repelled by the solid sheet. Novel morphological transitions between wrinkled, folded, and crumpled configurations are predicted to emerge at the borderlines between distinct parameter regimes.