The normalized $L^2$-norm of the traceless part of the Ricci curvature defines a Riemannian functional on the space of metrics. In this paper, we will consider this functional on 3-manifolds.

A group $G$ is said to enjoy ``Hasse principle'' if every local coboundary of $G$ is a global coboundary. Let $G$ be a non-Abelian finite $p$-group of order $p^m$, $p$ prime and $m > 4$ having a normal cyclic subgroup of order $p^{m-2}$ but having no element of order $p^{m-1}$. We prove that $G$ enjoys ``Hasse principle'' if $p$ is odd but in the case $p = 2$, there are fourteen such groups twelve of which enjoy ``Hasse principle'' but the remaining two do not satisfy ``Hasse principle''. We also find all the conjugacy preserving outer automorphisms for these two groups.

We classify the local structure of complex contact manifolds of dimension three with Legendrian vector fields. We also give the partial classification of the complex contact manifolds with Legendrian vector fields, in case the manifolds admit holomorphic contact 1-forms.