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**The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces.**
*(English)*
Zbl 0872.53035

Let \(M\) be a smooth connected manifold, \(G\) a complex or real linear algebraic group. It is known that deformations of a given representation \(\rho_0: \pi_1(M)\to G\) can be described in terms of flat connections in the vector bundle \(\operatorname{Ad} P\) associated to the flat principal \(G\)-bundle \(P\to M\) determined by \(\rho_0\). A deformation theory based on this idea is exposed in W. M. Goldman and J. J. Millson [Publ. Math., Inst. Hautes Etud. Sci. 67, 43-96 (1988; Zbl 0678.53059)]. In the present paper, a relative version of this theory is elaborated. Let \(U_1,\ldots,U_r\) be disjoint domains in \(M\), \(\Gamma_j\) the natural image of \(\pi_1(U_j)\) in \(\Gamma = \pi_1(M)\), and \(\mathcal O_j\) the \(\operatorname{Ad} G\)-orbit of \(\rho_0 |\Gamma_j\) in \(\operatorname{Hom}(\Gamma_j,G)\). Denote \(U = \bigcup_{j=1}^r U_j\) and \(R = \{\Gamma_1,\ldots,\Gamma_r\}\). The variety \(\operatorname{Hom}(\Gamma,R;G)\) of relative deformations of \(\rho_0\) is defined as the inverse image of \(\prod_{j=1}^r\mathcal O_j\) under the natural mapping \(\operatorname{Hom}(\Gamma,G)\to\prod_{j=1}^r \operatorname{Hom}(\Gamma_j,G)\). The authors construct a controlling differential graded Lie algebra \(\mathcal B(M,U;\operatorname{Ad} P)_0\) consisting of \(\operatorname{Ad} P\)-valued differential forms on \(M\); its fundamental property is that the complete local ring of \(\operatorname{Hom}(\Gamma,R;G)\) at the point \(\rho_0\) can be obtained from \(\mathcal B(M,U;\operatorname{Ad} P)_0\) by the procedure of the paper cited above.

This theory is applied to the study of deformations of mechanical linkages in one of the Riemannian spaces of constant curvature \(X = S^m, \mathbb{E}^m\) or \(\mathbb{H}^m\). One considers linkages \(\Lambda\) with \(n\) vertices \(u_1,\ldots,u_n\) such that any edge \(u_iu_j\) is the unique minimizing geodesic arc joining \(u_i\) and \(u_j\). The group \(G\) is the isometry group of \(X\), \(\Gamma\) is the free product \(\Phi_n\) of \(n\) copies of \(\mathbb{Z}/2\), \(\Gamma_j\) are dihedral subgroups corresponding to edges. Let \(\tau_1,\ldots,\tau_n\) be the generators of the \(\mathbb{Z}/2\) factors of \(\Phi_n\). Assigning to \(\tau_i\) the Cartan involution \(s_{u_i}\in G\) at the vertex \(u_i\), we get a representation \(\Phi_n\to G\). In this way, one obtains a local isomorphism between the configuration space of linkages with \(n\) vertices and \(\operatorname{Hom}(\Phi_n,R;G)\). Some special cases are considered and some problems are formulated.

This theory is applied to the study of deformations of mechanical linkages in one of the Riemannian spaces of constant curvature \(X = S^m, \mathbb{E}^m\) or \(\mathbb{H}^m\). One considers linkages \(\Lambda\) with \(n\) vertices \(u_1,\ldots,u_n\) such that any edge \(u_iu_j\) is the unique minimizing geodesic arc joining \(u_i\) and \(u_j\). The group \(G\) is the isometry group of \(X\), \(\Gamma\) is the free product \(\Phi_n\) of \(n\) copies of \(\mathbb{Z}/2\), \(\Gamma_j\) are dihedral subgroups corresponding to edges. Let \(\tau_1,\ldots,\tau_n\) be the generators of the \(\mathbb{Z}/2\) factors of \(\Phi_n\). Assigning to \(\tau_i\) the Cartan involution \(s_{u_i}\in G\) at the vertex \(u_i\), we get a representation \(\Phi_n\to G\). In this way, one obtains a local isomorphism between the configuration space of linkages with \(n\) vertices and \(\operatorname{Hom}(\Phi_n,R;G)\). Some special cases are considered and some problems are formulated.

Reviewer: A.L.Onishchik (Yaroslavl)

### MSC:

53C35 | Differential geometry of symmetric spaces |

70B15 | Kinematics of mechanisms and robots |

22E40 | Discrete subgroups of Lie groups |

57R22 | Topology of vector bundles and fiber bundles |

### Keywords:

relative deformation; differential graded Lie algebra; mechanical linkage; representation variety### Citations:

Zbl 0678.53059
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\textit{M. Kapovich} and \textit{J. J. Millson}, Compos. Math. 103, No. 3, 287--317 (1996; Zbl 0872.53035)

### References:

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