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We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n…

The rigidity theorem of Witten-Bott-Taubes-Hirzebruch tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then…

We construct a family of $6$-dimensional compact manifolds $M(A)$, which are simultaneously diffeomorphic to complex Calabi-Yau manifolds and symplectic Calabi-Yau manifolds. They have fundamental groups $\mathbb{Z} \oplus \mathbb{Z}$,…

Almost paracontact Riemannian manifolds of the lowest dimension are studied, whose paracontact distributions are equipped with an almost paracomplex structure. These manifolds are constructed as a product of a real line and a 2-dimensional…

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $\c{D}(M,N)$ is finite for any…

Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \to X$ from a complex torus $A.$ We prove that up to finite \'etale cover, $X$ is a product of projective spaces and a torus.

The formula for the Hirzebruch $\chi_y$-genus of complex manifolds is a consequence of the Hirzebruch-Riemann-Roch formula. The classical index formulae for Todd genus, Euler number, and Signature correspond to the case when the complex…

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds,…

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from…

We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection provided a metric is present. We rewrite the…

In our previous work, we introduced a special type of Hermitian metrics called {\em torsion-critical,} which are non-K\"ahler critical points of the $L^2$-norm of Chern torsion over the space of all Hermitian metrics with unit volume on a…

It is considered a differentiable manifold equipped with a pseudo-Riemannian metric and an almost contact 3-struc\-ture so that an almost contact metric structure and two almost contact B-metric structures are generated. There are…

Assume $(M, \omega)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

A toric manifold is a compact non-singular toric variety equipped with a natural half-dimensional compact torus action. A torus manifold is an oriented, closed, smooth manifold of dimension $2n$ with an effective action of a compact torus…

We introduce the notion of \emph{biharmonic almost complex structure} on a compact almost Hermitian manifold and we study its regularity and existence in dimension four. First we show that there always exist smooth energy-minimizing…

We determine all hyperbolic 3-manifolds $M$ admitting two toroidal Dehn fillings at distance 4 or 5. We show that if $M$ is a hyperbolic 3-manifold with a torus boundary component $T_0$, and $r,s$ are two slopes on $T_0$ with $\Delta(r,s) =…

For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the…

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. We also introduce some applications of this curvature identity.

A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with effective circular symmetry group of dimension 8 or less have positive Euler characteristic X(M): the fixed point set N consists of even…