  For functional data lying on an unknown nonlinear low-dimensional space, we
study manifold learning and introduce the notions of manifold mean, manifold
modes of functional variation and of functional manifold components. These
constitute nonlinear representations of functional data that complement
classical linear representations such as eigenfunctions and functional
principal components. Our manifold learning procedures borrow ideas from
existing nonlinear dimension reduction methods, which we modify to address
functional data settings. In simulations and applications, we study examples of
functional data which lie on a manifold and validate the superior behavior of
manifold mean and functional manifold components over traditional
cross-sectional mean and functional principal components. We also include
consistency proofs for our estimators under certain assumptions.
