  In this paper, we propose two methods for multivariate Hermite interpolation
of manifold-valued functions. On the one hand, we approach the problem via
computing suitable weighted Riemannian barycenters. To satisfy the conditions
for Hermite interpolation, the sampled derivative information is converted into
a condition on the derivatives of the associated weight functions. It turns out
that this requires the solution of linear systems of equations, but no vector
transport is necessary. This approach treats all given sample data points
equally and is intrinsic in the sense that it does not depend on local
coordinates or embeddings. As an alternative, we consider Hermite interpolation
in a tangent space. This is a straightforward approach, where one designated
point, for example one of the sample points or (one of) their center(s) of
mass, is chosen to act as the base point at which the tangent space is
attached. The remaining sampled locations and sampled derivatives are mapped to
said tangent space. This requires a vector transport between different tangent
spaces. The actual interpolation is then conducted via classical vector space
operations. The interpolant depends on the selected base point. The validity
and performance of both approaches is illustrated by means of numerical
examples.
