  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. The central issue is a differential geometric analysis of the
power flow solution space boundary (SSB, also in a simplifying way, called
saddle node bifurcation set, SNB) 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 points on curves in voltage or power space onto the SSB on points
closest to the given points on the curve, thus obtaining estimates for their
distance to the SSB. As another example how these concepts can be useful, we
present a new high precision continuation method for power flow solutions close
to and on the SSB called local inversion of the power flow map from voltage to
power space, assuming the dimension of power flow's Jacobean zero space, called
KERNEL, is one. For inversion, we present two different geometry-based
splitting techniques with one of them using the aforementioned orthogonal
tracing method.
