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It is well known that every 6-dimensional strictly nearly Kähler manifold $ (M,g,J)$0$ --> $ \operatorname{scal}>0$ of co-closed primitive $ (1,1)$ is stable under the Laplace operator $ \Delta$ $ E(\lambda)$-eigenspace of the restriction of $ \Delta$. If $ M$ $ \operatorname{scal}=30$ is naturally isomorphic to the direct sum $ E(2)\oplus E(6)\oplus E(12)$ [ABSTRACT FROM AUTHOR]

Abstract  We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal. [ABSTRACT FROM AUTHOR]

Abstract  In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with $${\mathbb{R}}$$ is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W 1 + W 4 in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds. [ABSTRACT FROM AUTHOR]

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics. [ABSTRACT FROM AUTHOR]

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-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular 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. [ABSTRACT FROM AUTHOR]