  A differential operator introduced by A. Gray on the unit sphere bundle of a
K\"ahler-Einstein manifold is studied. A lower bound for the first eigenvalue
of the Laplacian for the Sasaki metric on the unit sphere bundle of a
K\"ahler-Einstein manifold is derived. Some rigidity theorems classifying
complex space forms amongst compact Hermitian surfaces and the product of two
projective lines amongst all K\"ahler-Einstein surfaces are then derived.
