An n-body system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are n-configuration, configuration space, internal space, shape space, Jacobi transformation and weighted root system. The latter is a generalization of the root system of SU(n), which provides a bookkeeping for expressing the mutual distances of the point masses in terms of the Jacobi vectors. Moreover, its application to the study of collinear central n-configurations yields a simple proof of Moulton's enumeration formula. A major topic is the study of matrix spaces representing the shape space of n-body configurations in Euclidean k-space, the structure of the m-universal shape space and its O(m)-equivariant linear model. This also leads to those "orbital fibrations", where SO(m) or O(m) act on a sphere with a sphere as orbit space. A few of these examples are encountered in the literature, e.g. the special case S5/O(2) ≈ S4 was analyzed independently by Arnold, Kuiper and Massey in the 1970's.
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