In this paper we introduce the Cheeger-Gromoll type metric on coframe bundle of a Riemannian manifold and investigate Levi-Civita connection, curvature tensor, sectional geodesics with metric.

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C: the smoothing is a natural S-bundle over H, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S-bundle over H with symplectic...

We quantise complex, infinite–dimensional projective space CP(H). We apply the result to quantise a complex, finite–dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on C and the Fubini–Study metric on CP(H). Then the Hilbert–spa...

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is ηEinstein manifold if and only if base manifold is the space of constant sectional c...

Given a pair (semispray $S$, metric $g$) on a tangent bundle, the family of nonlinear connections $N$ such that $g$ is recurrent with respect to $(S, N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N, g)$ to be recurrent as well as for the triple $(S, stackrel{c}{N}, g)$ where $stackrel{c}{N}$ is the canonical ...

Given an effectively parameterized family of canonically polarized manifolds the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Applications concern the curvature of the classical and generalized Weil...

In this paper we relate the Fefferman–Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to...

We produce new examples of harmonic maps, having as either source or target manifold the tangent bundle TM on a Riemannian manifold (M,g), equipped with a Riemannian g-natural metric G. In particular, we study the harmonicity of the canonical projection π : (TM,G)→ (M,g), and of the identity map (TM,G)→ (TM,gS) and conversely, gS being the Sasaki metric on TM. A corresponding study is made for ...

Ray Singer torsion is a numerical invariant associated with a compact manifold equipped with a flat bundle, a Riemannian metric on the manifold and a Hermitian structure on the bundle. In this note we show how one can remove the dependence on the Riemannian metric and the Hermitian structure with the help of a base point and of an Euler structure, and obtain a topological invariant. A numerical...

In this article we continue studying the Ricci-flat Riemannian metrics that were constructed in [1]. On closer examination it turned out that they possess a number of remarkable properties; in particular, they have the holonomy group SU(2), so presenting special Kähler metrics. The metrics of holonomy SU(2) are interesting because of their applications in mathematical physics. In superstring th...