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In this paper, we investigate left invariant Riemannian metrics on Lie groups with one and two-dimensional commutator subgroups. We explicitly provide the Levi-Civita connection, sectional curvature, and Ricci curvature, and we give…

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of K\"ahler classes whose Bando-Calabi-Futaki…

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local…

In this paper, we establish a K\"ahlerian or projectivity criterion for a class of compact Hermitian surfaces with non-positive second Chern-Ricci curvature.

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by…

In this paper we consider left-invariant pseudo-K\"{a}hler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated…

We are concerned in this article with a classical topic in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of sectional curvature (resp. holomorphic sectional curvature) of a compact Riemannian…

It is well-known that every 6-dimensional strictly nearly K\"{a}hler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under…

We derive a canonical symmetry reduction associated to a compact non-K\"ahler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally K\"ahler, and we give a complete description in terms of a…

We classify six-dimensional homogeneous nearly K\"{a}hler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly K\"{a}hler manifold is a Riemannian 3-symmetric space equipped with its canonical almost…

A classical problem in constant mean curvature hypersurface theory is, for given $H\geq 0$, to determine whether a compact submanifold $\Gamma^{n-1}$ of codimension two in Euclidean space $\R_+^{n+1}$, having a single valued orthogonal…

One of the main purposes of this paper is to prove that on a complete K\"ahler manifold of dimension $m$, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum $\lambda_1(M) \ge m^2$, then it must…

We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the…

We are concerned in this article with a classical question in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of holomorphic sectional curvature of a complex $n$-dimensional compact K\"ahler…

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants…

The covariant derivative of the K\"ahler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can…

An anti-Kaehlerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd…

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is…

In this paper we study invariant almost Hermitian geometry on generalized flag manifolds which the isotropy representation decompose into two or three irreducible components. We will provide a classification of such flag manifolds admitting…

We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm.…