  In this paper, we are concerned with optimal control problems evolved on
Riemannian manifolds, where the initial and final states satisfy some
inequality and equality type constraints, and the control set is a separable
metric space. We obtain the second order necessary conditions of integral and
quasi-pointwise forms, both of which work for Pontryagin type critical controls
and involve the curvature tensor. Also, we apply the condition of integral form
to the Bolza problem, where the initial and final states are subject to
equality's type constraint.
