  This paper aims to systematically and comprehensively initiate a foundation
for using concepts from computational differential geometry as instruments for
power flow computing and research. At this point we focus our discussion on the
static case, with power flow equations given by quadratic functions defined on
voltage space with values in power space; both spaces have real Euclidean
coordinates. Central issue is a differential geometric analysis of the power
flow solution space boundary (SSB) both in voltage and in power space. We
present different methods for computing tangent vectors, tangent planes and
normals of the SSB and the normals' derivatives. Using the latter we compute
normal and principal curvatures. All this is needed for tracing the orthogonal
projection of curves in voltage and power space onto the SSB for points on the
SSB cosest to given points on the curves, thus obtaining estimates for the
distance to the SSB. Furthermore, we present a new high precision continuation
method for power flow solutions. We also compute geodesics on the SSB or an
implicitly defined submanfold thereof and, used to define geodesic coordinates
together with their Jacobians on the manifolds. These computations might be the
most innovative and most significant contribution of this paper, because this
concept provides a comprehensive coordinate system for sub many folds defined
by implicit equations. Therefore while moving on geodesics described by the
geodesic coordinates of the sub manifold at hand we get, via systematic
navigation guided by geodesic coordinates, access to all feasible operation
points of the system. We propose some applications and show some properties of
the Jacobian of the power flow map.
