Related papers: Strominger connection and pluriclosed metrics

We prove that there exist K\"{a}hler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that K\"{a}hler manifolds do not…

We study complex non-K\"ahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed…

Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a subgroup acting simply transitively on $M$ by hypercomplex isometries then the metric $g$ is…

In this paper, we consider general $k$th-mixed curvature $\mathcal{C}^{(k)}_{\alpha,\beta}$ ($\beta\neq0$) for Hermitian manifolds, which is a convex combination of the $k$th Chern Ricci curvature and holomorphic sectional curvature. We…

In this paper we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalizes the twistor space of quaternion-Hermitian manifolds and weak-Spin(9) structures. We also construct almost complex…

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere,…

A Hermitian metric on a complex manifold is called SKT (strong K\"ahler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called…

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety…

We investigate the geometry of a normal bundle equipped with a $(p,q)$-metric, i.e., Riemannian metric of Cheeger-Gromoll type, to the submanifold of a Riemannian manifold. We derive all natural object as the Levi-Civita connection,…

On an almost complex manifold, a quasi-K\"{a}hler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a non-degenerate solution of the c-projectively invariant metrizability…

A dualistic structure on a smooth Riemaniann manifold $M$ is a triple $(M,g,\nabla)$ with $g$ a Riemaniann metric and $\nabla$ an affine connection, generally assumed to be torsionless. From $g$ and $\nabla$, the dual connection $\nabla^*$…

In his recent work \cite{Y1}, X. Yang proved a conjecture raised by Yau in 1982 (\cite{Yau82}), which states that any compact K\"{a}hler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that…

We develop the moment map theory of the twisted scalar curvature of a K\"ahler metric. Primarily, we introduce a coupled system of equations on a holomorphic submersion intertwining the twisted scalar curvature of a K\"ahler metric on the…

We give a general construction of extremal Kaehler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kaehler…

We study Riemannian manifolds carrying a metric connection with parallel, skew-symmetric and closed torsion, which we call in short PSCT manifolds. We prove that PSCT manifolds always locally split into a product of well-understood factors,…

It is observed that on a compact almost complex Calabi-Yau with torsion (ACYT) 6-manifold with co-closed Lee form the curvature of the torsion connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the…

A vanishing theorem for uniformly RC $k$-positive Hermitian holomorphic vector bundles is established. It turns out that the holomorphic tangent bundle of a compact complex manifold equipped with a positive $k$-Ricci curvature K\"{a}hler…

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $\nabla^s = (1-\frac{s}{2})\nabla^c +…

In this paper, we systematically compute the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in…

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be…