In this paper, we study the statistical immersion of codimension one from a Sasakian statisticalmanifold of constant curvature to a holomorphic statistical manifold of constant holomorphiccurvature and its converse. We prove that in both cases the constant curvature equals to oneand the constant holomorphic curvature must be zero. Moreover, we construct several examplesof statistical manifolds, Sasakian statistical manifolds and holomorphic statistical manifolds ofconstant holomorphic curvature zero.

We prove vanishing theorems for the kernel of the Sampson Laplacian, acting on symmetric tensorson a Riemannian manifold and estimate its first eigenvalue on negatively pinched Riemannianmanifolds. Some applications of these results to conformal Killing tensors are presented.

We consider almost Einstein solitons (V; ) in a Riemannian manifold when V is a gradient, asolenoidal or a concircular vector field. We explicitly express the function by means of thegradient vector field V and illustrate the result with suitable examples. Moreover, we deduce somegeometric properties when the Ricci curvature tensor of the manifold satisfies certain symmetryconditions.

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler.A submanifold M of an almost Hermitian manifold is called a CR submanifold if it admits adifferentiable almost complex distribution D1 such that its orthogonal complement is a totally realdistribution. In this case the normal bundle of the submanifold also splits into two distributionsD3, which is almost complex, and a totally real complement. In the case of the proper threedimensionalCR submanifold of a sixdimensional manifold the distribution D3 is nontrivial.Here, we investigate threedimensional CR submanifolds of the sphere S6(1) admitting an umbilicdirection orthogonal to D3 and show that such submanifolds do not exist.

In this note, we characterize the fharmonic maps and bifharmonic maps with potential.We prove that every bifharmonic map with potential from complete Riemannian manifold,satisfying some conditions is a fharmonic map with potential. More, we study the case ofconformal maps between equidimensional manifolds.

In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle TMover an antiparaKähler manifold (M; '; g). We establish a necessary and sufficient conditionsunder which a covector field is harmonic with respect to the Berger type deformed Sasaki metric.We also construct some examples of harmonic vector fields. we also study the harmonicity of a mapbetween a Riemannian manifold and a cotangent bundle of another Riemannian manifold andvice versa.

In this paper, we study conics, which are invariant under the hyperbolic inversion with respectto the absolute of an extended hyperbolic plane H2 of curvature radius , 2 R+. They are calledthe hyperbolic Raisa Orbits of the second order.We prove that each hyperbolic Raisa Orbits of thesecond order in H2 belongs to one of four conics types of this plane. These types are as follows: thebihyperbolas of one sheet; the hyperbolas; the hyperbolic parabolas of one sheet and two branches;the elliptic cycles of radius =4. The family of all hyperbolic Raisa Orbits from the family of allbihyperbolas of one sheet (or all hyperbolas) defined exactly up to motions, is oneparametric. Thefamily of all hyperbolic Raisa Orbits from the family of all hyperbolic parabolas of one sheet andtwo branches (or all elliptic cycles) contains a unique conic defined exactly up to motions.

It was proved in Chen’s paper (Arch. Math. (Basel) 67 (1996), 519–528) that every real hypersurface
in the complex projective plane of constant holomorphic sectional curvature 4 satisfies
δ(2) ≤
9
4
H2 + 5,
where H is the mean curvature and δ(2) is a δinvariant introduced by him. In this paper, we study
nonHopf real hypersurfaces satisfying the equality case of the inequality under the condition that
the mean curvature is constant along each integral curve of the Reeb vector field. We describe how
to obtain all such hypersurfaces.

In the present paper, we study some notes on Berger type deformed Sasaki metric in the cotangent
bundle T
∗M over an antiparaKähler manifold (M, ϕ, g). We characterize some geodesic properties
for this metric. Next we also construct some almost antiparaHermitian structures on T
∗M and
search conditions for these structures to be antiparaKähler and quasiantiparaKähler with respect
to the Berger type deformed Sasaki metric.

In the present paper, we have studied some properties of a semisymmetric nonmetric connection
in HSUunified structure manifold and HSUKahler manifold. Some new results on such
manifolds have been obtained.
