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Rigid transformations groups. (English) Zbl 0652.53023
Géométrie différentielle, Colloq. Géom. Phys., Paris/Fr. 1986, Trav. Cours 33, 65-139 (1988).
[For the entire collection see Zbl 0635.00010.]
The group of diffeomorphisms that preserve a given geometric structure on a smooth manifold is an important object in differential geometry. This group is called the isometry group of the structure.
Many of the common geometric structures, for example Riemannian structures, projective structures, etc., can be described as a section of an appropriate fiber bundle associated to a natural principal bundle of jets over the manifold. It is this general type of geometric structure that is considered in this paper. Of these structures, the rigid structures are the main concern. In rough terms, a rigid structure is one where the local isometries are uniquely determined by derivatives of a sufficiently high order at a point. Thus a Riemannian structure is rigid, but a symplectic structure is not. (It should be noted in passing that the rigid structures generalize Cartan’s G-structures of finite type.)
After disposing of preliminaries such as the fact that an isometry group of a rigid geometric structure is a Lie group, the author then presents a deep and penetrating investigation into the structure of the groups that act as isometries of some rigid geometric structure.
Reviewer: J.Hebda

MSC:
53C10 \(G\)-structures
57S20 Noncompact Lie groups of transformations
53C99 Global differential geometry