## 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

Zbl 0678.53059
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### References:

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