On the solvability of the transvection group of extrinsic symplectic symmetric spaces

Let M$M$ be a symplectic symmetric space, and let ?:M→V$?:M\to V$ be an extrinsic symplectic symmetric immersion in the sense of Krantz and Schwachhöfer (2010) [7], i.e., (V,Ω)$(V,\Omega )$ is a symplectic vector space and ?$?$ is an injective symplectic immersion such that for each point p∈M$p\in M$, the geodesic symmetry in p$p$ is compatible with the reflection in the affine normal space at ?(p)$?\left(p\right)$.     

Uniqueness of balanced metrics on holomorphic vector bundles

Let E→M$E\to M$ be a holomorphic vector bundle over a compact Kähler manifold (M,ω)$(M,\omega )$. We prove that if E$E$ admits a ω$\omega$-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω$\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω$\omega$-balanced Kähler maps into Grassmannians.     

On the parallel transport of the Ricci curvatures

Geometrical characterizations are given for the tensor R⋅S$R\cdot S$, where S$S$ is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)$(M,g)$ and R$R$ denotes the curvature operator   acting on S$S$ as a derivation, and of the Ricci Tachibana tensor  _∧g⋅S${\wedge }_g\cdot S$, where the natural metrical operator  _∧g${\wedge }_g$ also acts as a derivation on S$S$. As a combination, the Ricci curvatures   associated with directions on M$M$, of which the isotropy determines that M$M$ is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on M$M$, and of which the isotropy determines that M$M$ is Ricci pseudo-symmetric in the sense of Deszcz.