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Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$., 16 pages

We describe and to some extent characterize a new family of K\"ahler spin manifolds admitting non-trivial imaginary K\"ahlerian Killing spinors., Comment: 23 pages, no figure

We prove that compact quaternionic-K\"ahler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces., Comment: the related manuscript arXiv:1006.2457 was merged into this new version

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo--Riemannian quaternion--Kaehler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that non--regular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds.

In this note, we look at estimates for the scalar curvatureof a Riemannian manifold M which are related to spin c Dirac operators: We show that one may not enlarge a Kahler metric with positive Ricci curvature without makingsmaller somewhere on M. We also give explicit upper bounds for minfor arbitrary Riemannian metrics on certain submanifolds of complex projective space. In certain cases, these estimates are sharp: we give examples where equality is obtained.

We consider the Dirac operator on compact quaternionic Kahler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kahler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kahler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.

In this paper Kahlerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kahler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kahlerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kahlerian Killing spinors are obtained.

We show that every Killing p-form on a compact quaternion-K\"ahler manifold has to be parallel for p greater than 1., Comment: erratum included