Dimitry V. Alekseevsky, Stefano Marchiafava
Quaternionic Transformations and the First Eigenvalues of Laplacian on a Quaternionic Kaehler Manifold
The paper is published: as "Transformations of a quaternionic Kaehler manifold", C. R. Acad. Sci., Paris, Ser. I 320, No.6, (1995) 703-708

MSC:

53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)

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

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Abstract: We consider quaternionic transformations of a quaternionic K\"ahler
manifold $(M,g,Q)$. General conditions for unicity of the
quaternionic K\"ahler structure $(g,Q)$ for given $Q$ are applied to
the case of two quaternionic K\"ahler metrics which are in
correspondence through a quaternionic transformation.
Characterization of compact quaternionic K\"ahler manifolds which
admit a quaternionic transformation $\ph$ different from an isometry
is given. A sharp estimate of first non zero eigenvalue for Laplacian
on a compact quaternionic K\"ahler manifold with positive scalar
curvature is also obtained, improving a previous result of C\.M\.
Margerin. An analogous sharp estimate for Dirac operator was
established by O\. Hijazi and J\.-L\. Milhorat, \cite 9.

Keywords: quaternionic manifold, automorphisms of G-structures, quaternionic vector field, Kaehler manifold eigenvalues of the Laplacian