  Various applications ranging from robotics to climate science require
modeling signals on non-Euclidean domains, such as the sphere. Gaussian process
models on manifolds have recently been proposed for such tasks, in particular
when uncertainty quantification is needed. In the manifold setting,
vector-valued signals can behave very differently from scalar-valued ones, with
much of the progress so far focused on modeling the latter. The former,
however, are crucial for many applications, such as modeling wind speeds or
force fields of unknown dynamical systems. In this paper, we propose novel
Gaussian process models for vector-valued signals on manifolds that are
intrinsically defined and account for the geometry of the space in
consideration. We provide computational primitives needed to deploy the
resulting Hodge-Mat\'ern Gaussian vector fields on the two-dimensional sphere
and the hypertori. Further, we highlight two generalization directions:
discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie
groups, and homogeneous spaces. Finally, we show that our Gaussian vector
fields constitute considerably more refined inductive biases than the extrinsic
fields proposed before.
