Abstract
We determine a 2-codimensional CR-structure on the slit tangent bundle \(T_0M\) of a Finsler manifold (M, F) by imposing a condition on the almost complex structure \(\Psi \) associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.
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References
M. Anastasiei, A framed \(f\)-structure on tangent manifold of a Finsler space. An. Univ. Bucur. Mat. Inf. 49(2), 3–9 (2000)
A. Bejancu, Geometry of CR-submanifolds Mathematics and its Applications, East European Series, vol. 23 (D. Reidel Publishing Co., Dordrecht, 1986)
A. Bejancu, H.R. Farran, Foliations and Geometric Structures Mathematics and Its Applications, vol. 580 (Springer, Dordrecht, 2006)
I. Bucataru, Z. Muzsnay, Projective and Finsler metrizability: parameterization-rigidity of the geodesics. Intern. J. Math. 23(9), 1250099 (2012), p. 15
S.-S. Chern, Z. Shen, Riemann–Finsler Geometry Nankai Tracts in Mathematics, vol. 6 (World Scientific Publishing Co. Pte. Ltd., Hackensack, 2005)
M. Crasmareanu, L.I. Piscoran, Para-CR structures of codimension 2 on tangent bundles in Riemann–Finsler geometry. Acta Math. Sin. (Engl. Ser.) 30(11), 1877–1884 (2014)
S. Dragomir, G. Tomassini, Differential Geometry and Analysis on CR Manifolds Progress in Mathematics, vol. 246 (Birkhäuser Boston Inc, Boston, 2006)
C. Ida, Some framed \(f\)-structures on transversally Finsler foliations. Ann. Univ. Mariae Curie-Sklodowska Sect. A 65(1), 87–96 (2011)
S.-Y. Kim, D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension. Topology 44(3), 557–584 (2005)
C. Medori, M. Nacinovich, Standard CR manifolds of codimension 2. Transform. Groups 6(1), 53–78 (2001)
R.I. Mizner, CR structures of codimension 2. J. Differ. Geom. 30(1), 167–190 (1989)
R.I. Mizner, Almost CR structures, \(f\)-structures, almost product structures and associated connections. Rocky Mt J. Math. 23(4), 1337–1359 (1993)
M.I. Munteanu, CR-Structures of CR-Codimension 2 on Hypersurfaces in Sasakian Manifolds, in Differential Geometry and Its Applications (Matfyzpress, Prague, 2005)
M.-I. Munteanu, New aspects on CR-structures of codimension 2 on hypersurfaces of Sasakian manifolds. Arch. Math. (Brno) 42(1), 69–84 (2006)
E. Peyghan, C. Zhong, A framed \(f\)-structure on the tangent bundle of a Finsler manifold. Ann. Pol. Math. 104(1), 23–41 (2012)
K. Yano, M. Kon, Structures on Manifolds Series in Pure Mathematics, vol. 3 (World Scientific Publishing Co., Singapore, 1984)
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The authors are thankfully to the referee(s) for several useful remarks which improve substantially the presentation and the contents of this paper.
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Crasmareanu, M., Pişcoran, LI. CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry. Period Math Hung 73, 240–250 (2016). https://doi.org/10.1007/s10998-016-0141-4
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DOI: https://doi.org/10.1007/s10998-016-0141-4