INVARIANT SUBMANIFOLDS OF CONTACT (Κ, Μ)-Manifolds

Abstract

Dedicated to the memory of Late Professor Jerzy J. Konderak on the second anniversary of his departure.Abstract. Invariant submanifolds of contact (κ, μ)-manifolds are studied. Our main result is that any invariant submanifold of a non-Sasakian contact (κ, μ)-manifold is always totally geodesic and, conversely, every totally geodesic submanifold of a nonSasakian contact (κ, μ)-manifold, μ = 0, such that the characteristic vector field is tangent to the submanifold is invariant. Some consequences of these results are then discussed.2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C25, 53D10, 53D15.

Submanifolds in Manifolds with Metric Mixed 3-Structures

“…2n+2 (R), tangent to the structure vector fields, is locally isometric with R 4k+3 2k+1 , R 4k+3 2k+2 , S 4k+3 2k+1 , S 4k+3 2k+2 , P 4k+3 2k+1 (R) and P 4k+3 2k+2 (R) respectively, where 0 ≤ k ≤ n. Proposition 5.2 and Corollary 3.10 together imply the following result, which corresponds to a theorem of Cappelletti Montano, Di Terlizzi and Tripathi [5] for submanifolds in contact (κ, µ)-manifolds. Remark 5.6.…”

Submanifolds of a (k,μ)-Contact Manifold

“…Motivated by these studies of the above authors [2,13,16], in the present paper we find the necessary and sufficient conditions for the submanifolds to be invariant and anti-invariant. Also we study CR-submanifolds of (k, µ)-contact manifold and examine the integrability of the horizontal and vertical distributions involved in the definition of CR-submanifolds of (k, µ)-contact manifold.…”

“…The study of submanifolds of different contact manifolds is carried out from 1970 onwards by several authors, for example [9]- [12], while the study of submanifolds of (k, µ)-contact manifold have been done by Montano et al [13], Avjit Sarkar et al [1], Tripathi et al [16], Siddesha and Bagewadi [14] and others.…”

“…In [13], the authors have shown that invariant submanifolds of (k, µ)-contact manifold carries a (k, µ) structure and prove the totally geodesicity of invariant submanifolds when the second fundamental form is parallel. Later authors of [2] and [15] continued the work of above authors and they proved the totally geodesicity of recurrent, generalized recurrent of second fundamental form and semiparallel, pseudoparallel, Ricci-generalized pseudoparallel submanifolds.…”

A classification of totally geodesic and totally umbilical Legendrian submanifolds of $$(\kappa ,\mu )$$ ( κ , μ ) -spaces

“…Nevertheless, the theory of submanifolds of (κ, µ)-spaces has not been developed in depth yet, even if we can find some very interesting papers about it. For example, in [4], B. Cappelletti Montano, L. Di Terlizzi and M. M. Tripathi proved that any invariant submanifold of a non-Sasakian contact (κ, µ)-space is always totally geodesic and, conversely, that every totally geodesic submanifold of a non-Sasakian contact (κ, µ)-space such that µ = 0 and the characteristic vector field ξ is tangent to the submanifold is invariant. Motivated by these results, we consider the case of submanifolds which are normal to ξ.…”