  We theoretically investigate properties of magnetostatic traps for cold atoms
that are subject to externally applied uniform fields. We show that Ioffe
Pritchard traps and other stationary points of $B$ are confined to a
two-dimensional curved manifold defined by $\det(\partial B_i/\partial x_j)=0$.
We describe how stationary points can be moved over the manifold by applying
external uniform fields. The manifold also plays an important role in the
behavior of points of zero field. Field zeroes occur in two distinct types, in
separate regions of space divided by the manifold. Pairs of zeroes of opposite
type can be created or annihilated on the manifold. Finally, we give examples
of the manifold for cases of practical interest.
