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Zbl 0872.53035
Kapovich, Michael; Millson, John J.
The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces. (English)
[J] Compos. Math. 103, No.3, 287-317 (1996). ISSN 0010-437X; ISSN 1570-5846/e

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 {\it W. M. Goldman} and {\it 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 $\Cal O_j$ 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 $\prod_{j=1}^r\Cal O_j$ 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 $\Cal B(M,U;\operatorname{Ad} P)_0$ 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 $\Cal B(M,U;\operatorname{Ad} P)_0$ by the procedure of the paper cited above. \par This theory is applied to the study of deformations of mechanical linkages in one of the Riemannian spaces of constant curvature $X = S^m, \bbfE^m$ or $\bbfH^m$. 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 $\bbfZ/2$, $\Gamma_j$ are dihedral subgroups corresponding to edges. Let $\tau_1,\ldots,\tau_n$ be the generators of the $\bbfZ/2$ 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.
[A.L.Onishchik (Yaroslavl)]

MSC 2000:: *53C35 Symmetric spaces (differential geometry)
70B15 Mechanisms
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

Cited in: Zbl 0989.53053 Zbl 0961.32026

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