  The classical Cartan's structural equations show in a compact way the
relation between a connection and its curvature, and reveals their geometric
interpretation in terms of moving frames. In order to study the mathematical
properties of singularities, we need to study the geometry of manifolds endowed
on the tangent bundle with a symmetric bilinear form which is allowed to become
degenerate. But if the fundamental tensor is allowed to be degenerate, there
are some obstructions in constructing the geometric objects normally associated
to the fundamental tensor. Also, local orthonormal frames and coframes no
longer exist, as well as the metric connection and its curvature operator. This
article shows that, if the fundamental tensor is radical stationary, we can
construct in a canonical way geometric objects, determined only by the
fundamental form, similar to the connection and curvature forms of Cartan. In
particular, if the fundamental tensor is non-degenerate, we obtain the usual
connection and curvature forms of Cartan. We write analogs of Cartan's first
and second structural equations. As a byproduct we will find a compact version
of the Koszul formula.
