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A submersion from an almost contact Riemannian manifold to an almost Hermitian manifold, acting on the horizontal distribution by preserving both the metric and the structure, is, roughly speaking a contact-complex Riemannian submersion. This paper deals mainly with a contact-complex Riemannian submersion from an η-Ricci soliton; it studies when the base manifold is Einstein on one side and when the fibres are η-Einstein submanifolds on the other side. Some results concerning the potential are also obtained here.

In this paper, an optimal inequality involving the delta curvature is exposed. With the help of this inequality some characterizations about the vertical motion and the horizontal divergence are obtained.

In this paper, we first introduce a new notion [Formula: see text]-tensor on Hermitian manifold and particularly, we present some geometric characterizations of such a tensor on the Kaehler manifold. Here, we investigate the Kaehler submersion whose total space is equipped with the [Formula: see text]-tensor and obtain some results. Also, we deal with a Kaehler submersion with totally geodesic fibers such that the total space admits [Formula: see text]-Ricci soliton and [Formula: see text]-tensor. Finally, we give necessary conditions for which any fiber and base manifold of Kaehler submersion is [Formula: see text]-Ricci soliton or [Formula: see text]-Kaehler-Einstein.

In this paper, we study the Riemannian submersions $\pi:M\rightarrow B$ whose total manifolds admit an almost Yamabe soliton. Here, we give some necessary conditions for which any fiber of $\pi$ or $B$ are almost Yamabe soliton or Yamabe soliton. Also, we calculate the scalar curvatures of any fiber or $B$ and using them, we present the relations between the scalar curvatures of them and obtain some characterizations of such a soliton (that is, shrinking, steady or expanding).

The classical notion of gradient Ricci soliton is extended here to the gradient Weyl-Ricci soliton. A Weyl structure of the base manifold M is lifted to its tangent bundle TM, by using the Sasaki metric. We give some necessary and sufficient conditions such that the Weyl structure on TM to be a gradient Weyl-Ricci soliton.

Two different notions of almost paracontact structures (which are compatible or anti-compatible with the metric), well known in the literature, are unified and generalized here. Several formulas of paraholomorphic maps are established, and a result of Lichnerowicz type is obtained. The connection transformations which have the same system of paracontact-planar Legendre curves are characterized. Conformal changes of metrics which preserve geodesics (resp. paracontact-planar Legendre curves) are studied.

In this paper, we consider the submanifold $M$ of a Kenmotsu manifold $\tilde M$ endowed with torqued vector field $\mathcal{T}$. Also, we study the submanifold $M$ admitting a Ricci soliton of both Kenmotsu manifold $\tilde M$ and Kenmotsu space form $\tilde M(c)$. Indeed, we provide some necessary conditions for which such a submanifold $M$ is an $\eta-$Einstein. We have presented some related results and classified. Finally, we obtain an important characterization which classifies the submanifold $M$ admitting a Ricci soliton of Kenmotsu space form $\tilde M(c)$.

In the present paper, we deal with the generic submanifold admitting a Ricci soliton in Sasakian manifold endowed with concurrent vector field. Here, we find that there exists never any concurrent vector field on the invariant distribution D of generic submanifold M. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D^{⊥} of M are Einstein. Finally, we give a characterization for a generic submanifold of Sasakian manifold to be a gradient Ricci soliton.

We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by Garcia-Rio and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Aahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O'Neill's book and Falcitelli, Ianus and Pastore's book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu's book to obtain some characterizations and to give a semi-Riemannian version of Fischer's (resp. Aahin's) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.

In this paper, we study Riemannian submersions whose total manifolds admit a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we study the harmonicity of Riemannian submersion from Ricci soliton and give a characterization for such a submersion to be harmonic.

On the total space of the cotangent bundle T* M of a Riemannian manifold (M, h) we consider the natural Riemann extension (g) over bar with respect to the Levi-Civita connection of h. In this setting, we construct on T double dagger M a new para-complex structure P, whose harmonicity with respect to (g) over bar is characterized here by using the reduction of (g) over bar to the (classical) Riemann extension.

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion [Formula: see text] has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.