It is a classical fact that the cotangent bundle T M of a differentiable manifold M enjoys a canonical symplectic form Ω. If (M, J, g, ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J̃, g̃,Ω), where Ω is the pull-back by g of Ω and g̃ is a pseudoRiemannian metric with neutral signature ...

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space formMm(c), c 6= 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

Let (M,ω) be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective has ample canonical bundle. In this paper, we show any irreducible subvariety of general type, thus confirming in particular case celebrated conjecture Lang. Moreover, can extend the theorem to quasinegative curvature building on earlier...

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form M̃2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form M̃2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively.

We obtain a conceptually new differential geometric proof of P.F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature kg(z) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ C approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂Ω.

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space Teich(Sg) of a closed surface Sg of genus g is uniformly bounded away...