Search
Submit an article
Differential Geometry of Manifolds

2020 №51

Back to the list Download the article
Bukusheva A. V.

Lifting semi-invariant submanifolds to distribu­tion of almost contact metric manifolds

DOI
10.5922/0321-4796-2020-51-5
Pages
39-48
almost contact metric manifold; section of a distribution; semi-invariant manifold; prolonged almost contact metric structure

Abstract

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection  that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection  is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section  of the distribution D defines a morphism  of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and  is a covariantly constant vec­tor field with respect to the N-connection , then  is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

Reference

1. Bukusheva, A. V.: On the Schouten — Wagner tensor of a nonholonomic Kenmotsu manifold. Proceedings of the seminar on geometry
and mathematical modeling, 5, 15—19 (2019).
2. Bukusheva, A.V.: Geodesic submanifolds of distributions of subRiemannian manifolds. Mathematics and science. Theory and practice.
Interuniversity collection of scientific papers. Yaroslavl. 23—27 (2019).
3. Bukusheva, A. V., Galaev, S.V.: On an admissible Kähler structure on a tangent bundle to a nonholonomic manifold. Mathematics. Mechanic, 7, 12—14 (2005).
4. 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., 12:3, 17—22 (2012).
5. Galaev, S.V.: Almost contact metric spaces with N-connection.Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15:3, 258—263 (2015).
6. Galaev, S.V.: Extended Structures on Codistributions of Contact Metric Manifolds. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.,17:2, 138—147 (2017).
7. Yampolsky, A. L.: On completely geodesic vector fields on a submanifold. Mat. physical, analysis, geometry, 1:1-2, 540—545 (1996).
8. Bejancu, A.: Geometry of CR-Submanifolds. Springer (1986).
9. Bukusheva, A.V., Galaev, S.V.: Almost contact metric structures defined by connection over distribution. Bull. of the Transilvania University of Brasov, Ser. III: Mathematics, Informatics, Physics, 4 (53):2, 13—22 (2011).
10. Walczak, P.: On totally geodesic submanifolds of tangent bundle with Sasaki metric. Bull. Acad. Polon. Sci., Ser. Sci. Math., 28:3-4, 161—165 (1989).