  We prove two results related to the Schwarz lemma in complex geometry. First,
we show that if the inequality in the Schwarz lemmata of Yau, Royden and
Tosatti becomes equality at one point, then the equality holds on the whole
manifold. In particular, the holomorphic map is totally geodesic and has
constant rank. In the second part, we study the holomorphic sectional curvature
on an almost Hermitian manifold and establish a Schwarz lemma in terms of
holomorphic sectional curvatures in almost Hermitian setting.
