Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.     

On almost constant-type manifolds

We introduce the class of almost constant-type manifolds and prove some theorems concerning Ricci tensors, scalar curvatures, bisectional curvatures and curvature identities. The above class is also studied in relation to other known classes of almost hermitian manifolds.To Adriano Barlotti with friendship and esteemThis work has been partially supported by a contribution of Ministero Ricerca Scientifica e Tecnologica.     

Homogeneous almost disjoint families

An almost disjoint family is constructed which is isomorphic to any almost disjoint family which can be constructed from it by taking subsets and finite unions. This is applied to the construction of a Boolean algebra with related properties.Presented by R. McKenzie.Partially supported by Israel-US Binational Research Fund.Partially supported by NSERC.     

Homogeneous Ricci almost solitons

It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or locally isometric to a product R×N(c), where N(c) is a space of constant curvature.     

Homogeneous and inhomogeneous manifolds

All metaLindelöf, and most countably paracompact, homogeneous manifolds are Hausdorff. Metacompact manifolds are never rigid. Every countable group can be realized as the group of autohomeomorphisms of a Lindelöf manifold. There is a rigid foliation of the plane.

     

Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

Lelong functional on almost complex manifolds

We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.     

Affine connections on almost para-cosymplectic manifolds

Identities for the curvature tensor of the Levi-Cività connection on an almost para-cosymplectic manifold are proved. Elements of harmonic theory for almost product structures are given and a Bochner-type formula for the leaves of the canonical foliation is established.     

On conformally flat almost Hermitian manifolds

We study some of 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor. Received 13 May 2000; revised 15 February 2001.     

Curvature of indefinite almost contact manifolds

The curvature tensor of indefinite almost contact manifolds is investigated. By means of the study of the Jacobi operator along spacelike, timelike and null geodesies, spaces of constant curvature are characterized as well as spaces of pointwise constant -sectional curvature. As an extension of these conditions we introduce the socalled -isotropic spaces and show a local classification of such manifolds.Supported by projects XUGA 20701B93 and DGICYT PB94 — 0633 — C02 — 01     

Chern numbers of almost complex manifolds

It is shown that any system of numbers that can be realised as the system of Chern numbers of an almost complex manifold of dimension , , can also be realised in this way by a connected almost complex manifold. This answers an old question posed by Hirzebruch.

     

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.