In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant mean of totally real holomorphic spacelike submanifolds in statistical manifolds type para-Kähler space forms. Furthermore, investigate equality cases these inequalities. As illustrations applications above inequalities, consider a few examples.

The study of surfaces immersed in a 3-dimensional ambient space plays a central role in the theory of submanifolds. In addition, Riemannian manifolds with constant sectional curvature can be considered as the most simple examples. Thus, one can think of surfaces with constant Gauss curvature in the Euclidean space R, hyperbolic space H or 3-sphere S as very natural objects of study. Through the...

We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermit-ian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov–Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization theorem for the sphere. 1. Uniformization in genus 0 and holomorphic structures ...

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, rigidity and degeneracy theorems without assuming any pointwise curvature signs both the domain target manifolds are proved, in which key roles played by total integration of function first eigenvalue second Ricci an almost nonpositivity notion sectional introduced our previous work. also apply...

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. Using a Penrose transform we show that, when (M, Q) is compact and admits a compatible quaternionic-Kähler metric of negative scalar curvature, Θ admits no global non-trivial holomorphic sections with respect to any of its holomorphic structures...

It was proved in Chen's paper [3] that every real hypersurface the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$\delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is mean and $\delta(2)$ a $\delta$-invariant introduced by him. In this paper, we study non-Hopf hypersurfaces satisfying equality case inequality under condition along each integral curve Reeb vector ...