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Conformal mappings of $$G$$-structures. (English. Russian original) Zbl 0748.53015
Funct. Anal. Appl. 22, No. 4, 311-313 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 68-69 (1988).
Let $$G\subset GL(n,\mathbb{R})$$ be a connected unimodular linear group and $$\tilde G=\mathbb{R}^ +G$$ its conformal extension. Any $$G$$-structure $$\pi: P\to M$$ may be naturally extended to a $$\tilde G$$-structure $$\tilde \pi: P\to M$$. A group $$A$$ of conformal transformations of a $$G$$-structure $$\pi$$ (that is a group of automorphisms of $$\tilde\pi$$) is called nonessential, if it preserves some $$G$$-structure $$\pi_ 1$$ with the same conformal extension $$\tilde\pi_ 1=\tilde\pi$$. Theorem. A group $$A$$ of conformal transformations of a $$G$$-structure $$\pi: P\to M$$ is nonessential, if one of the following conditions is fulfilled: 1) the group $$A$$ has compact stationary subgroups and $$\pi$$ has finite order; 2) the group $$A$$ is reductive and acts on $$M$$ transitively, and $$M$$ is reductive as a homogeneous space of the group $$A$$.
Let $$G$$ be an irreducible linear group of finite type. The author determines $$G$$-structures, that have maximal groups of conformal transformations. These results may be applied to classical $$G$$-structures of second order, e.g. to conformal and quaternionic structures. In particular they give a new proof of the theorem about the classification of quaternionic structures with maximal automorphism group, see the author [Sov. Math., Dokl. 37, No. 2, 381-384 (1988); translation from Dokl. Akad. Nauk SSSR 299, No. 3, 521-525 (1988; Zbl 0695.17011)] and E. Musso [On the transformation group of a quaternionic manifold, Preprint 1989/90, Univ. of Florence “Ulisse Dini”].

##### MSC:
 53C10 $$G$$-structures 57S20 Noncompact Lie groups of transformations 53C20 Global Riemannian geometry, including pinching
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