We address the problem of estimating optimal curves for interpolation, smoothing, and prediction of values along
partially-observed stochastic processes. In particular, we focus on processes that evolve on certain nonlinear manifolds
of importance in computer vision applications. The observations are given as a set of time-indexed points on
manifolds denoting noisy observations of the process at those times. Fitted curves on manifolds amount to geometrically
meaningful and efficiently computable splines on manifolds. We adopt the framework developed in Samir et al.
(2010) that develops a Palais metric-based steepest-decent algorithm applied to the weighted sum of a fitting-related
and a regularity-related cost function. Using the rotation group, the space of positive-definite matrices, and KendallŐs
shape space as three representative manifolds, we develop the proposed algorithm for curve fitting. This algorithm
requires expressions for exponential maps, inverse exponential maps, parallel transport of tangents, and curvature
tensors on the chosen manifolds. These ideas are illustrated using a large number of experimental results on both
simulated and real data