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 S^{5}/O(*2*) ≈ S^{4} was analyzed
independently by Arnold, Kuiper and Massey in the 1970's.

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