  Kohonen's Self-Organizing Maps (SOMs) have proven to be a successful
data-reduction method to identify the intrinsic lower-dimensional sub-manifold
of a data set that is scattered in the higher-dimensional feature space.
Motivated by the possibly non-Euclidian nature of the feature space and of the
intrinsic geometry of the data set, we extend the definition of classic SOMs to
obtain the General Riemannian SOM (GRiSOM). We additionally provide an
implementation as a proof-of-concept for geometries with constant curvature. We
furthermore perform the analytic and numerical analysis of the stability limits
of certain (GRi)SOM configurations covering the different possible regular
tessellation of the map space in each geometry. A deviation between the
numerical and analytic stability limit has been observed for the square and
hexagonal Euclidean maps for very small neighbourhoods in the map space as well
as agreement in case of longer-ranged relations between the map nodes.
