AN OPTIMAL INEQUALITY FOR WARPED PRODUCT LIGHTLIKE SUBMANIFOLDS

( Sangeet Kumar ),( Megha Pruthi ) 호남수학회 2021 호남수학학술지 Vol.43 No.2

In this paper, we establish several geometric characterizations focusing on the relationship between the squared norm of the second fundamental form and the warping function of SCR-lightlike warped product submanifolds in an indefinite Kaehler manifold. In particular, we find an estimate for the squared norm of the second fundamental form h in terms of the Hessian of the warping function λ for SCR-lightlike warped product submanifolds of an indefinite complex space form. Consequently, we derive an optimal inequality, namely ||h||² ≥ 2q{Δ(lnλ) + ||∇(lnλ)||² + c 2 p}, for SCR-lightlike warped product submanifolds in an indefinite complex space form. We also provide one non-trivial example for this class of warped products in an indefinite Kaehler manifold.

CLASSIFICATION OF TWISTED PRODUCT LIGHTLIKE SUBMANIFOLDS

Sangeet Kumar,Megha Pruthi Korean Mathematical Society 2023 대한수학회보 Vol.60 No.4

In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M_⊥×_ϕM_T, where M_⊥ is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type M_T×_ϕM_⊥ of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.

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.

Screen generic lightlike submersions

Sangeet Kumar,Gaurav Sharma,Dinesh Kumar Sharma 호남수학회 2023 호남수학학술지 Vol.45 No.4

We introduce the study of a new class of a lightlike submersion ϕ : K1 → K′ from a screen generic lightlike submanifold of an indefinite Kaehler manifold K2 onto an indefinite almost Hermitian manifold K′, and show that for this case K′ must be an indefinite Kaehler manifold. Then, we derive a relationship between the holomorphic sectional curvatures of K2 and K′ . Finally, we present a classification theorem for a screen generic lightlike submersion, giving the relationship between the sectional curvatures of the total space K2 and the fibers.

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.

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.

A NOTE ON GCR-LIGHTLIKE WARPED PRODUCT SUBMANIFOLDS IN INDEFINITE KAEHLER MANIFOLDS

Kumar, Sangeet,Pruthi, Megha Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.4

We prove the non-existence of warped product GCR-lightlike submanifolds of the type K_⊥ × _λ K_T such that K_T is a holomorphic submanifold and K_⊥ is a totally real submanifold in an indefinite Kaehler manifold $\tilde{K}$. Further, the existence of GCR-lightlike warped product submanifolds of the type K_T × _λ K_⊥ is obtained by establishing a characterization theorem in terms of the shape operator and the warping function in an indefinite Kaehler manifold. Consequently, we find some necessary and sufficient conditions for an isometrically immersed GCR-lightlike submanifold in an indefinite Kaehler manifold to be a GCR-lightlike warped product, in terms of the canonical structures f and ω. Moreover, we also derive a geometric estimate for the second fundamental form of GCR-lightlike warped product submanifolds, in terms of the Hessian of the warping function λ.

SLANT LIGHTLIKE SUBMANIFOLDS OF INDEFINITE NEARLY KAEHLER MANIFOLDS

( Tejinder Kumar ),( Sangeet Kumar ),( Pankaj Kumar ) 호남수학회 2021 호남수학학술지 Vol.43 No.2

In the present paper, we introduce the study of slant lightlike submanifolds of indefinite nearly Kaehler manifolds. After proving some geometric results for the existence of slant lightlike submanifolds of indefinite nearly Kaehler manifolds, we give a non-trivial example of this class of lightlike submanifolds. Then, we derive some conditions for the integrability of the distributions associated with slant lightlike submanifolds of indefinite nearly Kaehler manifolds. Consequently, we study totally umbilical slant lightlike submanifolds of indefinite nearly Kaehler manifolds. Subsequently, we investigate minimal slant lightlike submanifolds of indefinite nearly Kaehler manifolds.

A Hybrid Novel Approach of Video Watermarking

Manoj Kumar,Sangeet Sriastava,Arnold Hensman 보안공학연구지원센터 2016 International Journal of Signal Processing, Image Vol.9 No.10

The rapid evaluation of image manipulation and tampering has evoked the need for protecting digital data. Therefore a copyright protection method is used to protect the user information known as watermarking. Watermarking is the major image processing technique used to authenticate digital data. In this paper a hybrid approach is proposed to protect the digital data and also authenticates the user information. The proposed approach is an improvement over traditional video watermarking approaches. It uses fractions from the watermarked image to embed into the unique frames of the original video. The whole approach is divided into three steps which are described in the next sections of this paper. To check the resilience and robustness of this approach, various attacks are implemented over the watermarked video. After attacks, watermark image is extracted from the watermarked video to check the robustness of this approach. The quality of the recovered watermarked image is analysed using bit error rate. Experimental results show that this approach is very robust compared to other approaches.