We define a global horizontal 4-form on a quaternionic contact manifold and show that it is closed if and only if the torsion of the Biquard connection vanishes provided the dimension is grater than seven. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3-sasakian.

We study half lightlike submanifold M of an indefinite transSasakian manifold such that its structure vector field is tangent to M . First we study the general theory for such half lightlike submanifolds. Next we prove some characterization theorems for half lightlike submanifolds of an indefinite generalized Sasakian space form.

The object of the present paper is to characterize Cotton tensor on a 3-dimensional Sasakian manifold admitting $\eta$-Ricci solitons. After introduction, we study manifolds and introduce new notion, namely, pseudo-symmetric manifolds. Next deal with 3-manifold Among others prove that such constant scalar curvature Einstein some appropriate conditions. Also, classify nature soliton metric. Fina...

A quasi–regular Sasakian structure on a manifold L is equivalent to writing L as the unit circle subbundle of a holomorphic Seifert C-bundle over a complex algebraic orbifold (X,∆ = ∑ (1− 1 mi )Di). The Sasakian structure is called positive if the orbifold first Chern class c1(X) − ∆ = −(KX + ∆) is positive. We are especially interested in the case when the Riemannian metric part of the Sasakia...

A Riemannian manifold (M, g) is semi-symmetric if (R(X,Y ) ◦ R)(U, V,W ) = 0. It is called pseudo-symmetric if R ◦ R = F, F being a given function of X, . . . ,W and g. It is called partially pseudosymmetric if this last relation is fulfilled by not all values of X, . . . ,W . Such manifolds were investigated by several mathematicians: I.Z. Szabó, S. Tanno, K. Nomizu, R. Deszcz and others. In t...

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M̃ with the deck transform group acting conformally on M̃ . If M admits a holomorphic flow, acting on M̃ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifo...

It is proved that a Riemannian manifold M isometrically immersed in a Sasakian space form˜M(c) of constant ϕ-sectional curvature c < 1, with the structure vector field ξ tangent to M, satisfies Chen's basic equality if and only if it is a 3-dimensional minimal invariant submanifold. 1. Introduction. Let˜M be an m-dimensional almost contact manifold endowed with an almost contact structure (ϕ,ξ,...

A complete solution to the quaternionic contact Yamabe equation on qc sphere of dimension $4n+3$ as well Heisenberg group is given. uniqueness theorem for problem in a compact locally 3-Sasakian manifold shown.