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Galaev A. S. Classification of Prolonged Bi-metric Structures on Distributions of Non-zero Curvature of Sub-Riemannian Manifolds . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 3, pp. 263-273. DOI:


Classification of Prolonged Bi-metric Structures on Distributions of Non-zero Curvature of Sub-Riemannian Manifolds


The notion of the interior geometry of a sub-Riemannian manifold M is introduced, that is the aggregate of those manifold properties that depend only on the framing D ⊥ of the distribution D of the sub-Riemannian manifold as well as on the parallel transport of the vectors tangent to the distribution D along the curves tangent to this distribution.The maininvariantsof the interiorgeometry of a sub-Riemannianmanifold M are the following: the Schouten curvature tensor; the 1-form η defining the distribution D; the Lie derivative L ~
ξ g ofthemetrictensorg alongavectorfield ~ ξ;thetensorfieldPthatwithrespecttoadaptivecoordinateshasthe components P c ad = ∂ n Γ c ad . Depending on the properties of these invariants, 12 classes of sub-Riemannian manifolds are defined. Using the interior connection on the sub-Riemannian manifold M, an almost contact structure with a bi-metric is defined on the distribution D, which is called the prolonged structure in the paper. The comparison of two classifications of the prolonged structures is given. Accordance with the first classification, there are 12 classes of the prolonged structures corresponding to the 12 classes of the initial sub-Riemannian manifolds. The second classification is grounded on the properties of the fundamental F of type (0,3) associated with the bi-metrical structure. According to the second classification, there exist 2 11 classes of bi-metrical structures, among that 11 are basis classes F i , i = 1,...,11. The paper considers the case of a sub-Riemannian manifold with non-zero Schouten curvature tensor and with zero Lie derivative L ~ ξ g. It is proved that the prolonged almost contact bi-metrical structures corresponding to sub-Riemannian structures with the invariant ω = dη equal to zero, belong to the class F 1 ⊕ F 2 ⊕ F 3 , and the ones with non-zero invariant а ω = dη belong to the class F 1 ⊕ F 2 ⊕ F 3 ⊕ F 7 ⊕ ... ⊕ F 10 


1. Ganchev G., Mihova V., Gribachev K. Almost contact manifolds with B-metric // Math. Balkanica (N.S.). 1993. Vol. 7, fasc. 3–4. P. 261–276.
2. Manev M. Tangent bundles with Sasaki metric and almost hypercomplex pseudo-Hermitian structure // Topics in Almost Hermitian Geometry and Related Fields. 2005. P. 170–185. DOI:
3. Manev M. Tangent bundles with complete lift of the base metric and almost hypercomplex Hermitian-Norden structure // Compt. rend. Acad. bulg. Sci. 2014. Vol. 67, № 3. P. 313–322. arXiv:1309.0977v1 [math.DG].

4. Bukusheva A. V. The geometry of the contact metric spaces ϕ-connection. Belgorod State University Scientific Bulletin. Mathematics & Physics, 2015, no. 17(214), iss. 40, pp. 20– 24 (in Russian).
5. Bukusheva A. V., Galaev S. V. Connections on distributions and geodesic sprays. Russian Math. (Iz. VUZ), 2013, vol. 57, iss. 4, pp. 7–13. DOI:
6. Bukusheva A. V., Galaev S. V. Almost contact metric structures defined by connection over distribution with admissible Finslerian metric. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 3, pp. 17–22 (in Russian).
7. Galaev S. V. On classification of continuous B-metric structures on sub-Riemannian manifolds with zero Schouten tensor. Vestnik Bashkirskogo universiteta, 2017, vol. 22, no. 4, pp. 936–939 (in Russian).
8. Bukusheva A. V., Galaev S. V. Almost contact metric structures defined by connection over distribution. Bulletin of the Transilvania University of Brasov. Ser. III: Mathematics, Informatics, Physics, 2011, vol. 4 (53), no. 2, pp. 13–22.
9. Galaev S. V. Smooth distributions with admissible hypercomplex pseudo-hermitian structure. Vestnik Bashkirskogo Universiteta, 2016, vol. 21, no. 3, pp. 551–555 (in Russian).
10. Bukusheva A. V. Foliation on distribution with Finslerian metric. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 3, pp. 247–251 (in Russian).
11. Galaev S. V. On Distributions with Special Quasi-Sasakian Structure. Science Journal of VolSU. Mathematics. Physics, 2017, no. 2(39), pp. 6–17 (in Russian). DOI:
12. Vershik A. M., Gershkovich V. Ya. Nonholonomic dynamical systems. Geometry of distributions and variational problems. Dynamical systems–7, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow, VINITI, 1987. Vol. 16, pp. 5–85 (in Russian).
13. Vershik A. M., Faddeev L. D. Lagrangian mechanics in invariant formulation. Selecta Math. Soviet, 1981, vol. 1, no. 4, pp. 339–350. Reprinted in: L. D. Faddeev, 40 years in mathematical physics.
14. Gladush V. D. Five-Dimensional General Relativity and Kaluza–Klein Theory. Theoret. and Math. Phys., 2003, vol. 136, no. 3, pp. 1312–1324. DOI:
15. Manin Yu. I. Kalibrovochnye polia i kompleksnaia geometriia [Calibration fields and complex geometry]. Moscow, Nauka, 1984. 336 p. (in Russian).
16. Sasaki S. On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J., 1962, no. 14, pp. 146–155.
17. Yano K., Ishihara S. Tangent and cotangent bundles: differential geometry. New York, Marcel Dekker, Inc., 1973. 423 p.
18. Vagner V. V. Geometriia (n − 1)-mernogo negolonomnogo mnogoobraziia v n-mernom prostranstve [The geometry of an (n − 1)-dimensional non-holonomic manifold in an n- dimensional space]. Tr. seminara po vektornomu i tenzornomu analizu [Proceedings of the seminar on vector and tensor analysis]. Moscow, Moscow Univ. Press, 1941, iss. 5, pp. 173–255 (in Russian).
19. Blair D. E. Contact manifolds in Riemannian geometry. Berlin, New York, Springer-Verlag, 1976. 146 p.

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