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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”].

53C10 \(G\)-structures
57S20 Noncompact Lie groups of transformations
53C20 Global Riemannian geometry, including pinching
Full Text: DOI
[1] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1964). · Zbl 0129.13102
[2] D. V. Alekseevskii, Mat. Sb.,89, 280-296 (1972).
[3] S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York?Heidelberg (1972). · Zbl 0246.53031
[4] S. Kobayashi and T. Nagano, J. Math. Mech.,13, 875-907 (1964).
[5] T. Ochiai, Trans. Math. Soc.,152, 159-193 (1970). · doi:10.1090/S0002-9947-1970-0284936-6
[6] N. Tanaka, J. Math. Soc. Jpn.,17, 103-139 (1965). · Zbl 0132.16303 · doi:10.2969/jmsj/01720103
[7] D. Alekseevskii (Alekseevski), Ann. Global Anal. Geometry,3, 59-84 (1985). · Zbl 0564.53029 · doi:10.1007/BF00054491
[8] A. E. Tymanov, Sov. Probl. Mat. Fund. Napr. VINITI,9, 225-246 (1986).
[9] D. V. Alekseevskii, Mat. Entsikl.,5, 249-254 (1985).
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