  Manifold-valued functional data analysis (FDA) recently becomes an active
area of research motivated by the raising availability of trajectories or
longitudinal data observed on non-linear manifolds. The challenges of analyzing
such data come from many aspects, including infinite dimensionality and
nonlinearity, as well as time-domain or phase variability. In this paper, we
study the amplitude part of manifold-valued functions on $\mathbb{S}^2$, which
is invariant to random time warping or re-parameterization. Utilizing the nice
geometry of $\mathbb{S}^2$, we develop a set of efficient and accurate tools
for temporal alignment of functions, geodesic computing, and sample mean
calculation. At the heart of these tools, they rely on gradient descent
algorithms with carefully derived gradients. We show the advantages of these
newly developed tools over its competitors with extensive simulations and real
data and demonstrate the importance of considering the amplitude part of
functions instead of mixing it with phase variability in manifold-valued FDA.
