An almost Hermitian
manifold whose almost complex structure is Killing is called a nearly Kaehler
manifold; the usual almost complex structure on the six-sphere is a wellknown
example.
The purpose of this note is to introduce the study of almost contact metric
manifolds whose almost contact structure tensors are Killing. In particular if such a
structure is normal it is cosymplectic. Hypersurfaces of nearly Kaehler manifolds are
also studied. As an example, it is shown that the five-sphere carries a nonnormal
almost contact metric structure. More generally, the induced structure on a compact
orientable hypersurface of a nearly Kaehler manifold of positive curvature cannot be
cosymplectic.
