Abstract The natural symmetries of Riemannian manifolds are described by the its Riemann curvature tensor. In that sense, most symmetric constant sectional ones. Its generalizations locally manifolds, semisymmetric and pseudosymmetric manifolds. analogous holomorphic Kaehler holomorphically Do they differ in some way from their analogues? Yes, we prove can all be characterized only terms planes...

In 1985, Amari [1] introduced an interesting manifold, i.e., statistical manifold in the context of information geometry. The geometry of such manifolds includes the notion of dual connections, called conjugate connections in affine geometry, it is closely related to affine geometry. A statistical structure is a generalization of a Hessian one, it connects Hessian geometry. In the present paper...

We derive some consequences of the Liouville theorem for plurisubharmonic functions L.-F. Tam and author. The first result provides a nonlinear version complex splitting (which splits off factor ℂ isometrically from simply connected Kähler manifold with nonnegative bisectional curvature linear growth holomorphic function) second set results concerns so-called k-hyperbolicity its connection nega...

Some observations about the local and global generality of gradient Kähler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincaré coordinates in the case that the Ricci curvature is positive and the vector fie...

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a “convex combination” of Ricci and holomorphic sectional curvature. We show that in case, the canonical line bundle nef. Moreover, if negative at some point, then manifold projective being big If addition negative, ample. As an application, answer question Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100...

It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a gr...

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. In particular, we show that any self-dual quaternionic connection D of (M,Q) induces an holomorphic structure ∂̄ on Θ. We construct a Penrose transform which identifies solutions of the Penrose operator P on (M,Q) defined by D with the space of ∂̄-...