  It is shown that many results, previously believed to be properties of the
Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon
connections. We prove the existence of $t$--Gauduchon Ricci-flat metrics on the
suspension of a compact Sasaki--Einstein manifold, for all $t \in (-\infty,1)$;
in particular, for the Bismut, Minimal, and Hermitian conformal connection. A
monotonicity theorem is obtained for the Gauduchon holomorphic sectional
curvature, illustrating a maximality property for the Chern connection and
furnishing insight into known phenomena concerning hyperbolicity and the
existence of rational curves. Moreover, we show a rigidity result for Hermitian
metrics which have a pair of Gauduchon holomorphic sectional curvatures that
are equal, elucidating a duality implicit in the recent work of Chen--Nie.
