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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.

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

Citations:

Zbl 0678.53059
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References:

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