Dimitry V. Alekseevsky, Stefano Marchiafava
Gradient Quaternionic Vector Fields and a Characterization of the Quaternionic Projective Space
Preprint series: ESI preprints

MSC:

53B35 Hermitian and Kahlerian structures, See also {32Cxx}

53C55 Hermitian and Kahlerian manifolds, See also {32Cxx}

Abstract: On a quaternionic K\"ahler manifold $(M^{4n},g,Q)$, of positive
reduced scalar curvature $\nu$, a gradient vector field $Z=gradf$
which preserves the quaternionic structure $Q$ is studied. The
corresponding potential $f$ is proved to be an eigenfunction of the
Laplacian with the eigenvalue $\mu=2\nu(n+1)$. A second order
differential equation for the 1-form $\xi=df$ is established. We
prove that this equation is equivalent to the Obata-Blair equation
for the pull-back $\Ps=\pi^*\xi$ of the 1-form $\xi$ on the total
space of the Sasakian $SO_3$-principal bundle $\pi:F\to M$ associated
with $(M^{4n},g,Q)$. Using the results of Obata, Blair and Ishihara
we characterize the quaternionic projective space as the unique
quaternionic K\"ahler manifold of positive scalar curvature which
satisfies one of the following properties: i) there exists a non
Killing vector field $Z$ which preserves $Q$: ii) there exists an
eigenfunction of the Laplacian with the eigenvalue $\mu$

Keywords: Quaternionic Kaehler manifold, Quaternionic vector field, eigenfunction of Laplacian