Abstract
We introduce the notion of contactly geodesic transformation of the metric of an almost-contact metric structure as a contact analog of holomorphically geodesic transformations of the metric of an almost-Hermitian structure. A series of invariants of such transformations is obtained. We prove that such transformations preserve the normality property of an almost-contact metric structure. We prove that cosymplectic and Sasakian manifolds, as well as Kenmotsu manifolds, do not admit nontrivial contactly geodesic transformations of the metric, which is a contact analog of the well-known result for Kählerian manifolds due to Westlake and Yano.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
T. Levi-Cività, “Sulle transformationi delle equazioni dinamiche,” Ann. Math. Milano, Ser. 2, 24 (1894), 255–300.
T. Y. Thomas, “On projective and equiprojective geometries of paths,” PWC Nat. Acad. Sci. USA, 11 (1925), 198–203.
H. Weyl, “Zur Infinitesimalgeometrie Einordnung der projectiven und der conformen Auffassung,” Göttingen Nachr. (1921), 99–112.
N. S. Sinyukov, Geodesic Maps of Riemannian Spaces [in Russian], Nauka, Moscow, 1979.
W. J. Westlake, “Hermitian spaces in geodesic correspondence,” Proc. Amer. Math. Soc., 5 (1954), no. 2, 301–303.
K. Yano, “Sur la correspondence projective entre deux espaces pseudohermitiens,” C. R. Acad. Sci. Paris, 239 (1956), 1346–1348.
V. F. Kirichenko, “The axiom of Φ-holomorphic planes in contact metric geometry,” Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), no. 4, 711–739.
V. F. Kirichenko, Differential-Geometric Structures on Manifolds [in Russian], Moscow State Pedagogical University, Moscow, 2003.
V. F. Kirichenko, “Generalized quasi-Kahlerian manifolds and axioms of CR-submanifolds in generalized Hermitian geometry, II,” Geom. Dedicata, 52 (1994), 53–85.
V. F. Kirichenko, “Methods of generalized Hermitian geometry in the theory of almost-contact manifolds.,” J. Soviet Math., 42 (1988)), no. 5, 1885–1919.
D. E. Blair, “Contact manifolds in Riemannian geometry,” Lecture Notes Math., 509 (1976), 1–145.
K. Kenmotsu, “A class of almost-contact Riemannian manifolds,” Tôhoku Math. J., 24 (1972), 93–103.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 80, no. 2, 2006, pp. 209–219.
Original Russian Text Copyright © 2006 by V. F. Kirichenko, N. N. Dondukova.
Rights and permissions
About this article
Cite this article
Kirichenko, V.F., Dondukova, N.N. Contactly geodesic transformations of almost-contact metric structures. Math Notes 80, 204–213 (2006). https://doi.org/10.1007/s11006-006-0129-0
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0129-0