  A new framework is developed to intrinsically analyze sparsely observed
Riemannian functional data. It features four innovative components: a
frame-independent covariance function, a smooth vector bundle termed covariance
vector bundle, a parallel transport and a smooth bundle metric on the
covariance vector bundle. The introduced intrinsic covariance function links
estimation of covariance structure to smoothing problems that involve raw
covariance observations derived from sparsely observed Riemannian functional
data, while the covariance vector bundle provides a rigorous mathematical
foundation for formulating such smoothing problems. The parallel transport and
the bundle metric together make it possible to measure fidelity of fit to the
covariance function. They also play a critical role in quantifying the quality
of estimators for the covariance function. As an illustration, based on the
proposed framework, we develop a local linear smoothing estimator for the
covariance function, analyze its theoretical properties, and provide numerical
demonstration via simulated and real datasets. The intrinsic feature of the
framework makes it applicable to not only Euclidean submanifolds but also
manifolds without a canonical ambient space.
