Related papers: On Hermitian manifolds whose Chern connection is A…

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of $t$--Gauduchon Ricci-flat metrics on the…

The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional…

In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds, and generalize the Aviles-McOwen's existence results [J. Differential Geom., 21 (1985): 269-281] from Poincar\'e…

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a…

Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of…

We show that, under the definiteness of holomorphic sectional curvature, the spaces of some holomorphic tensor fields on compact Chern-K\"{a}hler-like Hermitian manifolds are trivial. These can be viewed as counterparts to Bochner's…

We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order…

A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this…

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. A linking curve is defined as a curve in the tangent space whose composition with the exponential map is tree formed. This key idea is…

This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and…

We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures.

Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only…

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler…

The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.

Using mirror symmetry, we show that Chern-Simons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more…

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, a special…

In this paper we compare different notions of transversality for possible singular complex algebraic or analytic subsets of an ambient complex manifold and prove a refined intersection formula for their Chern-Schwartz-MacPherson classes. In…

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler when the constant is non-zero and must be Chern flat when the constant is zero. The…

In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities…

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.