Let M be a real hypersurface of complex space form n ( c ) $M^n(c)$ , ≠ 0 $c\ne 0$ . Suppose that the structure vector field ξ is an eigen Ricci tensor S, S = β $S\xi =\beta \xi$ being function. We study on M, gradient pseudo-Ricci soliton g f λ μ $M,g,f,\lambda ,\mu$ extended concept soliton, closely related to pseudo-Einstein hypersurfaces. When ≥ 3 $n\ge 3$ we show Hopf hypersurface.

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton’s equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometr...

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical manifold, which is conjugate Ricci-symmetric, has an equiaffine structure is giv...

In this paper, we study submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection on a submanifold is also semi-symmetric non-metric connection. We consider the total geodesicness and minimality of a submanifold with respect to the semi-symmetric non-metric connection. We obtain the Gauss, Cadazzi, and Ricci equations for submanifold...

We investigate the local metrizability of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$, where $\beta$ is a closed null 1-form. show that such space Berwald type if and only (pseudo-)Riemannian $\alpha$ 1-form have very specific form in certain coordinates. In particular, when signature Lorentzian, belongs to subclass Kundt class generates corresponding congruence, this ge...

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of 2N dimensional Ricci flat (pseudo-) Riemannian geometries.

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive. Dedicated to the memory of Profess...

A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is s...