Abstract
We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with the symmetries.
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References
Altomani, A., Medori, C., Nacinovich, M.: On homogeneous and symmetric CR manifolds. Boll. Unione Mat. Ital. 3(2), 221–265 (2010)
Čap, A., Slovák, J.: Parabolic Geometries: Background and general theory. American Mathematical Society, Providence (2009)
Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). Erratum Acta Math. 150 no. 3–4 (1983) 297
Gregorovič, J.: General construction of symmetric parabolic structures. Differ. Geom. Appl. 30(5), 450–476 (2012)
Gregorovič, J., Zalabová, L.: Symmetric parabolic contact geometries and symmetric spaces. Transform. Groups 18(3), 711–737 (2013)
Gregorovič, J., Zalabová, L.: Notes on symmetric conformal geometries. Arch. Math. TOMUS 51(5), 287–296 (2015)
Gregorovič, J., Zalabová, L.: On automorphisms with natural tangent action on homogeneous parabolic geometries. J. Lie Theor. 25, 677–715 (2015)
Gregorovič, J., Zalabová, L.: Geometric properties of homogeneous parabolic geometries with generalized symmetries. Diff. Geom. Appl. 49, 388–422 (2016)
Gregorovič, J., Zalabová, L.: Local generalized symmetries and locally symmetric parabolic geometries. SIGMA 13, 032–33 (2017)
Kaup, W., Zaitsev, D.: On symmetric Cauchy–Riemann manifolds. Adv. Math. 149(2), 145–181 (2000)
Kruglikov, B., The, D.: The gap phenomenon in parabolic geometries. J. Reine Angew. Math. 723, 153–215 (2017)
Loos, O.: Spiegelungsräume und homogene symmetrische Räume. Math. Zeitschr. 99, 141–170 (1967)
Onishchik, A.: On compact Lie groups transitive on certain manifolds. Sov. Math. Dokl. 1, 12881291 (1961)
Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of complex variables. J. Math. Soc. Jpn. 14, 397–429 (1962)
Zalabová, L.: A non-homogeneous, symmetric contact projective structure. Cent. Eur. J. Math. 12(6), 879–886 (2014)
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First author supported by the project P29468 of the Austrian Science Fund (FWF). Second author supported by the Grant 17-01171S of the Czech Science Foundation (GAČR)
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Gregorovič, J., Zalabová, L. On Symmetric CR Geometries of Hypersurface Type. J Geom Anal 29, 3135–3159 (2019). https://doi.org/10.1007/s12220-018-00110-1
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DOI: https://doi.org/10.1007/s12220-018-00110-1