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Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds…

Let $(M,g)$ be an n-dimensional Riemannian manifold and $T^{*}M$ be its cotangent bundle equipped with a Riemannian metric of Cheeger Gromoll type which rescale the horizontal part by a nonzero differentiable function. The main purpose of…

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine…

In this article, we give completely new examples of embedded complex manifolds the germ of neighborhood of which is holomorphically equivalent to a germ of neighborhood of the zero section in its normal bundle. The first set of examples is…

We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity…

The study of embeddings of smooth manifolds into Euclidean and projective spaces has been for a long time an important area in topology. In this paper we obtain improvements of classical results on embeddings of smooth manifolds, focusing…

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold…

The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue…

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold $M$ of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe…

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian…

Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold.…

Given a compact quantizable pseudo-K\"ahler manifold $(M,\omega)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2\pi i\,\omega$. We shall show that the asymptotic expansion of the Bergman…

In this paper we describe an approach to complex Finsler metrics suitable to deal with global questions, and stressing the similarities between hermitian and complex Finsler metrics. Let $F$ be a smooth complex Finsler metric on a complex…

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…

We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions under which…

In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we…

We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space…

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…

This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented…

The structure of nearly K\"ahler manifolds was studied by Gray in several papers. More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a strict and complete nearly K\"ahler manifold is…