We consider strict and complete nearly Kähler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kähler is locally a Riemannian product of homogenous nearly Kähler spaces, twistor spaces over quaternionic Kähler manifolds and 6-dimensional nearly Kähler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kähler structure. Finally, we obtain a classification result for the homogenous case, reducing a conjecture of Wolf and Gray to its 6-dimensional form.

We study the space of nearly Kähler structures on compact 6-dimensional manifolds. In particular, we prove that the space of infinitesimal deformations of a strictly nearly Kähler structure (with scalar curvature scal) modulo the group of diffeomorphisms is isomorphic to the space of primitive coclosed (1, 1)-eigenforms of the Laplace operator for the eigenvalue 2 scal/5.

Let ∇ be a metric connection with totally skew-symmetric torsion T on a Riemannian manifold. Given a spinor field Ψ and a dilaton function Φ, the basic equations in type II B string theory areWe derive some relations between the length ||T|| 2 of the torsion form, the scalar curvature of ∇, the dilaton function Φ and the parameters a, b, µ. The main results deal with the divergence of the Ricci tensor Ric ∇ of the connection. In particular, if the supersymmetry Ψ is non-trivial and if the conditions (dΦ T) T = 0 , δ ∇ (dT) • Ψ = 0 hold, then the energy-momentum tensor is divergence-free. We show that the latter condition is satisfied in many examples constructed out of special geometries. A special case is a = b. Then the divergence of the energy-momentum tensor vanishes if and only if one condition δ ∇ (dT) • Ψ = 0 holds. Strong models (dT = 0) have this property, but there are examples with δ ∇ (dT) = 0 and δ ∇ (dT) • Ψ = 0.

Abstract. We study 6-dimensional nearly Kähler manifolds admitting a Killing vector field of unit length. In the compact case it is shown that up to a finite cover there is only one geometry possible, that of the 3-symmetric space S 3 × S 3 .

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topological obstructions to the existence of a geometrically 2-formal Kähler metric, at the level of the second cohomology group. A strong interaction with almost Kähler geometry is to be noted. In complex dimension 3, we list all the possible values of the second Betti number of a geometrically 2-formal Kähler metric.