The ENKF attempts to predict the trajectory of a particle by combining
uncertain measurements of the particle's state vector (x(t_2)) with uncertain
information from a previous state of the state vector (x(t_1)). The particle is
assumed to be a random variable undergoing an affine transformation from t_1 to
t_2. We wish to extend this method of prediction to particles travelling in a
Riemannian or even a more general manifold. It is not immediately clear how we
can extend the notion of an affine transformation on a manifold, but it can be
done. We consider the simplest case in which the particle undergoes a parallel
translation. In R^n, this corresponds to x(t_2) = x(t_1) + w, where w is some
vector in R^n. One can visualize this as a vector field over R^n, with all
vectors being parallel and equal to w, and each point x(t_1) being translated
along this vector field. In a Riemannian manifold, we can extend this notion of
parallel translation. This is actually one version of the Fundamental Theorem
of Riemannian Geometry. We will explain how this theorem relates to parallel
translation. We will also derive transformation laws for the mean and
covariance matrix of a random vector on a Riemannian Manifold. We will also
explain the concept of tensors, which has many applications in analysis, linear
algebra, and pde's.
