Related papers: On the normal bundle of Levi-flat real hypersurfac…

Let $M\stackrel\pi \arrow X$ be a principal elliptic fibration over a Kaehler base $X$. We assume that the Kaehler form on $X$ is lifted to an exact form on $M$ (such fibrations are called positive). Examples of these are regular Vaisman…

In this paper, we extend our investigation of the class of biconservative surfaces with non-constant mean curvature in 4-dimensional space forms $N^4(\epsilon)$. Specifically, we focus on biconservative surfaces with non-parallel normalized…

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not…

We classify all the $6$-dimensional unimodular Lie algebras $\mathfrak{g}$ admitting a complex structure with non-zero closed $(3,0)$-form. This gives rise to $6$-dimensional compact homogeneous spaces $M=\Gamma\backslash G$, where $\Gamma$…

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which…

We give an example of a compact Levi-flat CR 3-manifold with a positive-along-leaves CR line bundle which is not ample with respect to transversely infinitely differentiable CR sections. This example shows that we cannot improve the…

In this paper, we consider real hypersurfaces $M$ in $\Bbb C^3$ (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is…

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior…

We prove that a compact Riemannian manifold $M$ does not admit any non-trivial $m$-modified homothetic vector fields. In the corresponding case of an $m$-modified conformal vector field $V$, we establish an inequality that implies the…

We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of $CP^n$, $HP^n$ and $CaP^2$, and a family of lens…

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show…

In this paper, we establish a "pseudo-effective" version of the holonomy principle for compact K\"{a}hler manifolds with nonnegative holomorphic sectional curvature. As applications, we prove that if a compact complex manifold $M$ admits a…

A long-standing conjecture in non-K\"ahler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be K\"ahler when $c\neq 0$ and must be…

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural…

Let X be a compact manifold with boundary and let L^k be a high power of a hermitian holomorphic line bundle over X. When X has no boundary, Demailly's holomorphic Morse inequalities give asymptotic bounds on the dimensions on the Dolbeault…

In this paper, we investigate the existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles over rational homogeneous varieties. Using Cartan's highest weight theory, we establish an explicit algebraic…

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit…

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form…

Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian…

Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues…