Slant immersions in C5C_5-manifolds

Odd-dimensional non anti-invariant slant submanifolds of an α- Kenmotsu manifold are studied. We relate slant immersions into a Kähler manifold with suitable slant submanifolds of an α-Kenmotsu manifold. More generally, in the framework of Chinea-Gonzalez, we specify the type of the almost contact metric structure induced on a slant submanifold, then stating a local classification theorem. The case of austere immersions is discussed. This helps in proving a reduction theorem of the codimension. Finally, slant submanifolds which are generalized Sasakian space-forms are described

Special almost Hermitian geometry

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the \SUn(n)-structure. Furthermore, we apply the obtained results to almost hyperhermitian geometry. Thus, we show that the exterior derivatives of the three K{\"a}hler forms of an almost hyperhermitian manifold are sufficient to determine the three covariant derivatives of such forms, i.e., the three mentioned exterior derivatives determine the intrinsic torsion of the {\sl Sp}(n)-structure.Comment: 27 page

Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure

For a given complex n-fold M we present an explicit construction of all complex (n+1)-folds which are principal holomorphic T2-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kaehler SU(3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus.Comment: LaTeX, 17 pages, organization of the paper was changed, typos correcte

Almost Hermitian 6-Manifolds Revisited

A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly K\"ahler manifold is parallel. On the other side, any almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem and we describe almost hermitian $\mathrm{G}_1$-manifolds with parallel torsion form. In particular, among them there are only two types of $\mathcal{W}_3$-manifolds with a non-abelian holonomy group, namely twistor spaces of 4-dimensional self-dual Einstein manifolds and the invariant hermitian structure on the Lie group \mathrm{SL}(2, \C). Moreover, we classify all naturally reductive hermitian $\mathcal{W}_3$-manifolds with small isotropy group of the characteristic torsion.Comment: 26 pages, revised versio

Curvature decomposition of G_2 manifolds

Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a $G_2$ structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed $G_2$ manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one $G_2$ manifolds. As a consequence every Fern\'andez-Gray type of $G_2$ structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in $G_2$.Comment: LaTeX 2e, 26 pages, 2 tables. Changes in version 2: shortened, reorganized, misprints corrected, several remarks and new introduction. A formula in the proof of Theorem 1.2a has been corrected. Submitte

Riemannian submersions from almost contact metric manifolds

In this paper we obtain the structure equation of a contact-complex Riemannian submersion and give some applications of this equation in the study of almost cosymplectic manifolds with Kaehler fibres.Comment: Abh. Math. Semin. Univ. Hamb., to appea

The complex Goldberg-Sachs theorem in higher dimensions

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+\epsilon)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $\epsilon \in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of $\mathcal{N}$, and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from four to higher dimensions. Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to $2^m$ holomorphic distributions of totally null $m$-planes. Finally, we adapt these findings to the category of real smooth pseudo-Riemannian manifolds, commenting notably on the applications to Hermitian geometry and Robinson (or optical) geometry.Comment: Section 2 partly rewritten: issue regarding self-duality clarified. Section 5.2 clarified. Some remarks added. Lemma 3.7 (previously 3.7) corrected. A few mathematical and notational inaccuracies corrected, and typos and sign mistakes fixed throughout. Some references adde