We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kähler–Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show that any 4-dimensional pseudo-Hermitian manifold also admits a unique Kähler–Weyl structure.

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kähler covering M̃ , with the deck transform acting on M̃ by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its volume form. This result is a conformal analogue of Calabi’s theorem stating ...

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also relate the Cotton-York tensor of an umbilic hypersurface to...

In this paper, we study a special class of generalized Douglas-Weyl metrics whose Douglas curvature is constant along any Finslerian geodesic. We prove that for every Landsberg metric in this class of Finsler metrics, ? = 0 if and only if H = 0. Then we show that every Finsler metric of non-zero isotropic flag curvature in this class of metrics is a Riemannian if and only if ? = 0.

in this paper, we study a special class of generalized douglas-weyl metrics whose douglas curvature is constant along any finslerian geodesic. we prove that for every landsberg metric in this class of finsler metrics, ? = 0 if and only if h = 0. then we show that every finsler metric of non-zero isotropic flag curvature in this class of metrics is a riemannian if and only if ? = 0.

This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl’s theorems exist except for his conformal theorem in the 4-dimensiona...

in this paper, we study a class of finsler metrics which contains the class of p-reducible andgeneral relatively isotropic landsberg metrics, as special cases. we prove that on a compact finsler manifold,this class of metrics is nothing other than randers metrics. finally, we study this class of finsler metrics withscalar flag curvature and find a condition under which these metrics reduce to r...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type G2. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of t...

On a (pseudo-) Riemannian manifold of dimension n > 3, the space of tensors which transform covariantly under Weyl rescalings of the metric is built. This construction is related to a Weyl-covariant operator D whose commutator [D,D] gives the conformally invariant Weyl tensor plus the Cotton tensor. So-called generalized connections and their transformation laws under diffeomorphisms and Weyl r...