6 New paper on flat 2-torus frameworks.

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a linkage (e.g., a rectangle) whose edges are geodesics of fixed length, and whose vertices are joints, and again ask if it is rigid or flexible on the surface. This leads to my first, specific question:

Q1. Is a rhombus, or a rectangle, always flexible on a sphere?

It seems the answer should be Yes but I am a bit uncertain if there must be a restriction on the edge lengths. (In the above figure, the four arcs are each $49^\circ$ in length, comfortably short.)

Q2. The same question for other surfaces: Arbitrary convex surfaces? A torus?

I am especially interested to learn if there are situations where a linkage that is flexible in the plane is rendered rigid when embedded on some surface. It seems this should be possible...?

Q3. More generally, Laman's theorem provides a combinatorial characterization of the rigid linkages in the plane. The $n{=}4$ rectangle is not rigid because it has fewer than $2n-3 = 5$ bars: it needs a 5th diagonal bar to rigidify. Has Laman's theorem been extended to arbitary (closed, smooth) surfaces embedded in $\mathbb{R}^3$? Perhaps at least to spheres, or to all convex surfaces?

Thanks for any ideas or pointers to relevant literature!

Addendum. I found one paper related to my question: "Rigidity of Frameworks Supported on Surfaces" by A. Nixon, J.C. Owen, S.C. Power. arXiv:1009.3772v1 math.CO In it they prove an analog of Laman's theorem for the circular cylinder in $\mathbb{R}^3$. If one phrases Laman's theorem as requiring for rigidity that the number of edges $E \ge 2 V - 3$ in both the graph and in all its subgraphs, then their result (Thm. 5.3) is that, on the cylinder, rigidity requires $E \ge 2 V -2$ in the graph and in all its subgraphs. This is not the precise statement of their theorem. They must also insist that the graph be regular in a sense that depends on the rigidity matrix achieving maximal rank (Def. 3.3). They give as examples of irregular linkages on a sphere one that contains an edge with antipodal endpoints, or one that includes a triangle all three of whose vertices lie on a great circle. But modulo excluding irregular graphs and other minor technical details, they essentially replace the constant 3 in Laman's theorem for the plane with 2 for the cylinder.

Theirs is a very recent paper but contains few citations to related work on surfaces, suggesting that perhaps the area of linkages embedded on surfaces is not yet well explored. In light of this apparent paucity of information, it seems appropriate that I 'accept' one of the excellent answers received. Thanks!

Addendum [31Jan11]. I just learned of a 2010 paper by Justin Malestein and Louis Theran, "Generic combinatorial rigidity of periodic frameworks" arXiv:1008.1837v2 (math.CO), which pretty much completely solves the problem of linkages on a flat 2-torus, generalizing to flat orbifolds. They obtain a combinatorial characterization for generic minimal rigidity for "planar periodic frameworks," which encompass these surfaces.

5 Accept note.

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a linkage (e.g., a rectangle) whose edges are geodesics of fixed length, and whose vertices are joints, and again ask if it is rigid or flexible on the surface. This leads to my first, specific question:

Q1. Is a rhombus, or a rectangle, always flexible on a sphere?

It seems the answer should be Yes but I am a bit uncertain if there must be a restriction on the edge lengths. (In the above figure, the four arcs are each $49^\circ$ in length, comfortably short.)

Q2. The same question for other surfaces: Arbitrary convex surfaces? A torus?

I am especially interested to learn if there are situations where a linkage that is flexible in the plane is rendered rigid when embedded on some surface. It seems this should be possible...?

Q3. More generally, Laman's theorem provides a combinatorial characterization of the rigid linkages in the plane. The $n{=}4$ rectangle is not rigid because it has fewer than $2n-3 = 5$ bars: it needs a 5th diagonal bar to rigidify. Has Laman's theorem been extended to arbitary (closed, smooth) surfaces embedded in $\mathbb{R}^3$? Perhaps at least to spheres, or to all convex surfaces?

Thanks for any ideas or pointers to relevant literature!

Addendum. I found one paper related to my question: "Rigidity of Frameworks Supported on Surfaces" by A. Nixon, J.C. Owen, S.C. Power. arXiv:1009.3772v1 math.CO In it they prove an analog of Laman's theorem for the circular cylinder in $\mathbb{R}^3$. If one phrases Laman's theorem as requiring for rigidity that the number of edges $E \ge 2 V - 3$ in both the graph and in all its subgraphs, then their result (Thm. 5.3) is that, on the cylinder, rigidity requires $E \ge 2 V -2$ in the graph and in all its subgraphs. This is not the precise statement of their theorem. They must also insist that the graph be regular in a sense that depends on the rigidity matrix achieving maximal rank (Def. 3.3). They give as examples of irregular linkages on a sphere one that contains an edge with antipodal endpoints, or one that includes a triangle all three of whose vertices lie on a great circle. But modulo excluding irregular graphs and other minor technical details, they essentially replace the constant 3 in Laman's theorem for the plane with 2 for the cylinder.

Theirs is a very recent paper but contains few citations to related work on surfaces, suggesting that perhaps the area of linkages embedded on surfaces is not yet well explored. In light of this apparent paucity of information, it seems appropriate that I 'accept' one of the excellent answers received. Thanks!

Addendum.I found one paper related to my question:"Rigidity of Frameworks Supported on Surfaces"by A. Nixon, J.C. Owen, S.C. Power.arXiv:1009.3772v1 math.COIn it they prove an analog of Laman's theorem forthe circular cylinder in $\mathbb{R}^3$.If one phrases Laman's theorem as requiring forrigidity that the number of edges $E \ge 2 V - 3$in both the graph and in all its subgraphs,then their result (Thm. 5.3) is that, on the cylinder, rigidity requires$E \ge 2 V -2$ in the graph and in all its subgraphs.This is not the precise statement of their theorem.They must also insist that the graph be regularin a sense that depends on the rigidity matrix achieving maximal rank(Def. 3.3).They give as examples of irregular linkages on a spherecontains an edge with antipodal endpoints, or one that includesa triangle all three of whose vertices lie on a great circle.But modulo excluding irregular graphs and other minor technicaldetails, they essentially replace the constant 3 in Laman'stheorem for the plane with 2 for the cylinder.