  In this paper, we present a comprehensive proof concerning the regularity of
critical points for the spline energy functional on Riemannian manifolds, even
for the general higher-order case. Although this result is widely acknowledged
in the literature, a detailed proof was previously absent. Our proof relies on
a generalization of the DuBois-Reymond Lemma. Furthermore, we establish the
existence of minimizers for the spline energy functional in cases where
multiple interpolation points are prescribed alongside just one velocity.
