Contact CR-warped product submanifolds in cosymplectic space forms

Chen type inequality for warped product immersions in cosymplectic space forms

“…Similar inequalities have been done for other spaces (see [1,3]). In our research findings, the base manifold is considered to be an invariant submanifold of the ambient manifold.…”

“…for each j = n 1 + 1, · · · , n. The contact CR-warped product submanifolds of cosymplectic manifolds were studied in [1,5]. In this section, first we give the following lemma.…”

Semi-invariant warped product submanifolds of almost contact manifolds

“…Finally, if equality holds identically then from Equation ( 4 Semi-invariant warped product submanifolds of a generalized Sasakian space-form Hesigawa and Mihai [9] obtained the inequality for squared norm of second fundamental form for contact CR-warped product submanifolds in the setting of Sasakian space form. In the available literature, similar estimates are proved for squared norm of second fundamental form in contact manifolds (c.f., [3,4]). Since generalized Sasakian space form include the class of all almost contact metric manifold, so in this section we will obtain an estimate for the squared norm of second fundamental form for semi-invariant warped product submani-folds in the setting of generalized Sasakian space form.…”

“…With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on a warped product manifolds for instance and warped product manifolds are widely used in differential geometry, Physics and as well as in different branches of Engineering. Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research, and many articles are available in literature (c.f., [2][3][4]). CR-warped product was introduced by Chen [5].…”

“…Hesigawa and Mihai [9] obtained the inequality for squared norm of the second fundamental form in term of the warping function for contact CR-warped product in Sasakian manifolds. Recently Atceken [4] studied contact CR-warped product submanifolds in Cosymplectic space-forms and obtained an inequality for second fundamental form in terms of warping function. After reviewing the literature, we realized that there is very few studies on the warped product submanifold for almsot contact manifolds so it will be worthwhile to study the warped product submnifolds in the setting of almost contact metric manifold.…”