The origin of charge and internal energy remains one of the mysteries of
physics. In that which follows, techniques of differential geometry yield a
clue that charge density and internal energy density, like inertia and mass, may
have its roots in the curvature properties of implicit surfaces.

The differential geometry of implicit surfaces can be studied in terms of a map
from a variety x^{k} into a normal vector field, A_{k} which is
homogeneous of degree zero in its component functions. The Jacobian matrix of
the map is singular and will induce a metric on the variety which is also
singular. The similarity invariants of the singular Jacobian matrix yield
expressions in terms of the curvatures of the implicit surface. In the simplest
of cases, the normal vector field of the implicit surface may be used to
construct a 1-form, A=A_{k}dx^{k} which is integrable in the
sense of Frobenius. That is, the one-form may be reduced to the product of a
single function times the total differential (gradient field) of a second
function, A=a db. On the
otherhand it is possible to start from an arbitrary 1-form of Action potentials,
A_{0} which need not satisfy the Frobenius unique integrability theorem,
and thereby describe more complex implicit surfaces.

For if an arbitrary 1-form is divided by a suitable Holder norm l the component functions, A=A_{0}/l can be made homogeneous of degree zero, and most of the
results for the simple implicit surfaces also follow. Recall that A_{0}
may be used to formulate the exterior differential system F_{0 }-
dA_{0} = 0, which, when dA_{0 }is not zero, leads to the
Maxwell-Faraday equations as a system of constraints on the variety. These
observations imply a possible connection between electromagnetism and curvature
of implicit surfaces. The conjecture is further strengthened by noting that the
Adjoint matrix to the singular Jacobian matix can be used to construct a well
defined N-1 form density, J_{s. } This N-1 form has global zero
divergence, d J_{s} = 0. Hence, J_{s} - dG_{s} = 0,
which defines an exterior differential system equivalent to the Maxwell-Ampere
equations, and leads to concept that J_{s} represents an electromagnetic
current density.

For those Holder norms which are equivalent to the Gauss map, the interaction
between these closed currents J_{s} and the potentials, A, generated by
the N-form A^J_{s} has remarkable properties. For 2D surfaces in 3
dimensions, it has been observed that the interaction coefficient of
A^J_{s} is exactly equal to the Gauss curvature of the implicit surface
defined by the homogeneous 1-form, A. In any dimension, the N-form of
interaction A^J_{s} has a coefficient equal to the curvature similarity
invariant of degree N-1. For N = 4, this result implies that the charge current
density is cubic in the curvatures of the implicit surface. As gravity and mass
energy density are currently considered to be artifacts of quadratic (Gauss)
curvature, the conjecture follows that the energy density of interaction between
potentials and currents, as well as charge, may be artifacts of cubic (Adjoint)
curvature, and not Gauss sectional curvature. By using the known relationship
between A^J_{s} and the first Poincare invariant, it would appear that
Adjoint curvature is a specific gauge addition, d (A^G_{s}) to the
Lagrangian of the field, F^G_{s} This result raises the credence level
in the physical significance of what has been called topological spin,
A^G_{s} and suggest a possible role for studying the electromagnetic
interaction and modification or manipulation of the gravitational field.

By studying the evolution of the physical system described by the Action 1-form,
with respect to a vector field generated by the charge current density via Cartan's Magic formula , it also becomes
apparent that the Adjoint curvature and the "internal Energy" are formally the
same. Hence internal energy density, interaction energy density, and Adjoint
curvature of the implicit hypersurface generated by the 1 form of Action
Potentials are equivalent concepts.