  In this work we develop a novel and foundational framework for analyzing
general Riemannian functional data, in particular a new development of tensor
Hilbert spaces along curves on a manifold. Such spaces enable us to derive
Karhunen-Loeve expansion for Riemannian random processes. This framework also
features an approach to compare objects from different tensor Hilbert spaces,
which paves the way for asymptotic analysis in Riemannian functional data
analysis. Built upon intrinsic geometric concepts such as vector field,
Levi-Civita connection and parallel transport on Riemannian manifolds, the
developed framework applies to not only Euclidean submanifolds but also
manifolds without a natural ambient space. As applications of this framework,
we develop intrinsic Riemannian functional principal component analysis
(iRFPCA) and intrinsic Riemannian functional linear regression (iRFLR) that are
distinct from their traditional and ambient counterparts. We also provide
estimation procedures for iRFPCA and iRFLR, and investigate their asymptotic
properties within the intrinsic geometry. Numerical performance is illustrated
by simulated and real examples.
