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Riemannian submersions between Lie groups and Riemannian homogeneous spaces are investigated. With the help of connections, some characterizations dealing these spaces are obtained.

Almost semiinvariant ? -submanifolds of an almost paracontact metric manifold are dened and studied. Some characterizations of almost semiinvariant ? -submanifolds and semiinvariant ? -submanifolds are presented. A para-CR-structure is dened and it is proven that an almost semiinvariant ? -submanifold of a normal almost paracontact metric (and hence para-Sasakian) manifold with the proper invariant distribution always possesses a para-CR-structure. A counter example is also given. Integrability conditions for certain natural distributions arising on almost semiinvariant ? -submanifolds are obtained. Finally, certain parallel operators on submanifolds are investigated.

In this paper, we study some characterizations for any submanifold of a golden Riemannian manifold to be semi-invariant in terms of canonical structures on the submanifold, induced by the golden structure of the ambient manifold. Besides, we determine forms of the distributions involved in the characterizations of a semi-invariant submanifold on both its tangent and normal bundles.

Almost semiinvariant x\perp-submanifolds of an almost paracontact metric manifold are defined and studied. Some characterizations of almost semiinvariant x\perp-submanifolds and semiinvariant x\perp-submanifolds are presented. A para-CR-structure is defined and it is proven that an almost semiinvariant x\perp-submanifold of a normal almost paracontact metric (and hence para-Sasakian) manifold with the proper invariant distribution always possesses a para-\textit{CR}-structure. A counter example is also given. Integrability conditions for certain natural distributions arising on almost semiinvariant x\perp -submanifolds are obtained. Finally, certain parallel operators on submanifolds are investigated.

Biharmonic Hypersurfaces of LP-Sasakian Manifolds In this paper the biharmonic hypersurfaces of Lorentzian para-Sasakian manifolds are studied. We firstly find the biharmonic equation for a hypersurface which admits the characteristic vector field of the Lorentzian para-Sasakian as the normal vector field. We show that a biharmonic spacelike hypersurface of a Lorentzian para-Sasakian manifold with constant mean curvature is minimal. The biharmonicity condition for a hypersurface of a Lorentzian para-Sasakian manifold is investigated when the characteristic vector field belongs to the tangent hyperplane of the hypersurface. We find some necessary and sufficient conditions for a timelike hypersurface of a Lorentzian para-Sasakian manifold to be proper biharmonic. The nonexistence of proper biharmonic timelike hypersurfaces with constant mean curvature in a Ricci flat Lorentzian para-Sasakian manifold is proved.

In this paper, the area vectors of the closed spherical indicator of a unit dual sphere and the parallel projection areas of the planar regions surrounded in the lines space of these the closed spherical indicators are calculated in terms of the components of G, Koenigs screw. In addition, a orihonormal trihedron fıxed to the striction point X of an initial nıled surface and some relations were obtained for the parallel projection areas of the spherical indicator of this trihedron.

In this paper, J. Jacobi’s Theorems [9] have been considered for the spherical curves drawn on the unit dual sphere during the closed space motions. The integral invariants of the ruled surfaee corresponding, in the line space, to the spherical curve dtawn by a fixed point on the moving unit dual sphere during the one-parameter closed motion were calculated with a different approach from the area vector used by H. R. Müller [11], In addition, the ruled surfaces corresponding to the curves drawn by the unit tangent vector, principal normal vector or a unit vector on the osculating plane of the mentioned curve, were seen to be cones with this approach.

Dual quaternion-valued functions of a single real variable define a curve in ID4. The Serret-Frenet formulae for a curve in ID3, well known [1], it is rederived with the help of dual spatialquaternions. Making use of these formulae, we derived the Serret-Frenet formulae for dual quaternion-valued functions in ID4.

In this paper, the relation between the Steiner vector of a one-parameter closed spherical motion and the area vector of the closed spherical curve formed under the motion is discussed and some corollaries are given. Moreover, making use of the area formula of W. Blaschke and the area vector defined by H. R. Muller, the formula given by H. H. Hacisalihoǧlu is obtained by a different method.

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian paracosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian paracosymplectic manifold., Comment: This paper has been withdrawn by the author

In this paper, the co-screen conformal 1-lightlike submanifolds of a Lorentzian manifoldare introduced as a generalization of co-screen locally half-lightlike submanifolds in(Wang, Wang {\&} Liu, 2013; Wang {\&} Liu, 2013) and two examples are given whichone is co-screen locally conformal andthe other is not. Some results are obtained on these submanifolds whichthe co-screen distribution is conformal Killing on the ambient manifold.The induced Ricci tensor of co-screen conformal 1-lightlike submanifolds isinvestigated.

In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We investigate hypersurfaces of affinely cosymplectic and normal (1,1,1) almost contact manifolds. It is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. Some necessary and sufficient conditions have been given for a noninvariant hypersurface of a Lorentzian para-Sasakian manifold to be locally product manifold. We establish a Lorentzian para-Sasakian structure for an invariant hypersurface of a Lorentzian para-Sasakian manifold. Finally we give some examples for invariant and noninvariant hypersurfaces of a Lorentzian para-Sasakian manifold., Comment: 16 pages