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GCR-LIGHTLIKE SUBMANIFOLDS OF INDEFINITE NEARLY KAEHLER MANIFOLDS
Kumar, Sangeet,Kumar, Rakesh,Nagaich, R.K. Korean Mathematical Society 2013 대한수학회보 Vol.50 No.4
We introduce CR, SCR and GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds and obtain their existence in indefinite nearly Kaehler manifolds of constant holomorphic sectional curvature $c$ and of constant type ${\alpha}$. We also prove characterization theorems on the existence of totally umbilical and minimal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds.
Kaul Shreya,Jain Neha,Nagaich Upendra 한국약제학회 2022 Journal of Pharmaceutical Investigation Vol.52 No.2
Purpose The aim of this investigation was to formulate and evaluate naproxen transethosomal gel for sustained transdermal delivery for the management of musculoskeletal pain. Methods In this examination naproxen sodium-loaded transethosomes were developed by ethanol injection method. A 9 run, 2-factor, 3-level factorial design was used to optimize naproxen-transethosomes. Transethosomal formulations were then incorporated into hydrogel made of gelling agent carbopol 940. The formulated transethosomes were characterized for particle size, entrapment efficiency, zeta potential, in-vitro release, ex-vivo drug permeation study, drug deposition study, and in-vivo anti-inflammatory study. Results The results exhibited that the particle size were in the range of 56.94 ± 0.12 nm to 291.7 ± 0.09 nm. The transethosomes had higher entrapment efficiency in between 66.23 ± 1.52 and 93.11 ± 0.96% and exhibited a spherical morphology when examined by TEM analysis. The in-vitro skin permeation study carried out on rat skin exhibited enhanced skin deposition with lesser systemic absorption. The in-vivo studies carried out on rats showed the superiority of naproxen transethosomal gel in reducing the edema rate. Conclusion The results obtained all together demonstrated that the formulated transethosomal gel possessed a smaller particle size, high entrapment efficiency along higher skin deposition rate which is required in getting relief from musculoskeletal pain. The developed formulation could be regarded as an ideal substitute for the conventional gel for the management of musculoskeletal pain.
Integrability of Distributions in GCR-lightlike Submanifolds of Indefinite Sasakian Manifolds
Jain, Varun,Kumar, Rakesh,Nagaich, Rakesh Kumar Department of Mathematics 2013 Kyungpook mathematical journal Vol.53 No.2
In this paper, we study GCR-lightlike submanifolds of indefinite Sasakian manifold. We give some necessary and sufficient conditions on integrability of various distributions of GCR-lightlike submanifold of an indefinite Sasakian manifold. We also find the conditions for each leaf of holomorphic distribution and radical distribution is totally geodesic.
Submanifolds of Sasaki-like Almost Contact Manifolds with B-metric
Anu Devgan,Rakesh Kumar Nagaich 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.3
In this paper, we introduce the geometry of contact CR submanifolds and radical transversal lightlike submanifolds of Sasaki-like almost contact manifolds with Bmetric. We obtain some new results that establish a relationship between these two submanifolds.
GCR-LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN PRODUCT MANIFOLD
Kumar, Sangeet,Kumar, Rakesh,Nagaich, Rakesh Kumar Korean Mathematical Society 2014 대한수학회보 Vol.51 No.3
We introduce GCR-lightlike submanifold of a semi-Riemannian product manifold and give an example. We study geodesic GCR-lightlike submanifolds of a semi-Riemannian product manifold and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product. Finally, we discuss minimal GCR-lightlike submanifolds of a semi-Riemannian product manifold.
INTEGRABILITY OF DISTRIBUTIONS IN GCR-LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS
Kumar, Rakesh,Kumar, Sangeet,Nagaich, Rakesh Kumar Korean Mathematical Society 2012 대한수학회논문집 Vol.27 No.3
In present paper we establish conditions for the integrability of various distributions of GCR-lightlike submanifolds and obtain conditions for the distributions to define totally geodesic foliations in GCR-lightlike submanifolds.
GCR-LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN PRODUCT MANIFOLD
Sangeet Kumar,Rakesh Kumar,Rakesh Kumar Nagaich 대한수학회 2014 대한수학회보 Vol.51 No.3
We introduce GCR-lightlike submanifold of a semi-Riemann- ian product manifold and give an example. We study geodesic GCR- lightlike submanifolds of a semi-Riemannian product manifold and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product. Finally, we discuss minimal GCR-lightlike submanifolds of a semi-Riemannian product manifold.
GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds
Sangeet Kumar,Rakesh Kumar,R. K. Nagaich 대한수학회 2013 대한수학회보 Vol.50 No.4
We introduce CR, SCR and GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds and obtain their existence in indefinite nearly Kaehler manifolds of constant holomorphic sectional curvature c and of constant type α. We also prove characterization theorems on the existence of totally umbilical and minimal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds.