Rigidity of Symmetric Frameworks on the Cylinder

A bar-joint framework (<i>G</i>, <i>p</i>) is the combination of a finite simple graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and a placement <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo>→</mml:mo> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>d</mml:mi></mml:msup> </mml:mrow> </mml:math> . The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:math> restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.