The gamma ray field we live in is extremely rich and dense. For the forms we find in nuclei and assorted particles, there is all the energy needed to drive all physical processes.

A Calabi-Yau shape within a nucleus or particle needs an external energy supply to maintain it. Gravity provides the energy. Here we are talking about force and pressure within a nucleus or particle, with only indirect connection to the outside, or connection at a point, curve, or surface. There may also be tears joining and reforming.

Occasionally we refer to neutrons, protons, electrons, and nuclei. A proton can be a hydrogen nucleus, though we list it separately when we talk about free protons, such as in the solar wind, particle colliders, or elsewhere. Let’s take an Oxygen nucleus for example with the makings of 8 protons and 8 neutrons. Inside the nucleus, at the top, parachutes with baskets attached through ropes, or strings, instead of a parachutist, may cause some gravitons to loop around the insides of the parachutes, or branes, and into the baskets with enough force to hold the parachutes against the highest flux density of gravitons. Then the gravitons would find ways to tunnel through the baskets, pushed from behind. In the motions of O_{2} in air, the parachutes may slide around to stay opposite the maximum flux.

This may also help explain weak interaction parity violation, because as an electron forming within a nucleus tries to escape, out the bottom is easier, due to escape out the top involving going through the gaps in the parachutes. More than 50% would come out downward.

The manifold of the sun’s gamma ray field, the manifold of the earth’s gamma ray field, and likewise with other celestial bodies, provides a combination of symmetric spaces. During the day, at noon let’s say, the vectors of the sun’s manifold are in the opposite direction as the vectors of the earth’s terrestrial manifold. The Coulomb field uses all vectors of all manifolds to propagate, because all vectors, within a distance of 10 meters at least, are frozen in time for phonon transmission.

Let’s say M_{1} is the earth’s manifold, and M_{2} is the combination of the earth’s and sun’s manifolds. “…a diffeomorphism F: M_{1} → M_{2} of manifolds oriented by Ω_{1}, Ω_{2}, is orientation-preserving if F*Ω_{2} = λΩ_{1}, where λ > 0 is a C^{ꝏ} function on M.” ([1] pg. 209) In our example here, λ > 1, and we have neglected the earth’s moon for simplification.

We may call a negative charge a left coset space, and a positive charge a right coset space. Each creates its own homomorphism in the dense gamma ray field, by a diffeomorphism on the electric fields of the gamma rays. For one thing, there is circular polarization. For another, perpendicular to the greatest flux density of gamma rays the electric fields of the gamma rays may have skewed sine wave lobes, somewhere between a normal sine wave and a sawtooth. The Coulomb field acts tangent to the R vector sphere, and “(∇_{X}Y)_{p} depends not on the vector field X but only on its value X_{p} at p.” ([1] pg. 309] The way that the Coulomb field transmits radially is by centrifugal force through the gamma ray field.

The inside of an atom may be called a geodesic. An electron path in an atomic orbital may also be called a geodesic, and “a long geodesic may not be minimal.” ([2] pg. 62) This is due to the Lorentz force:

https://www.fruechtetheory.com/blog/2010/12/23/electron-orbitals-and-the-lorentz-force/

Gravity is an integral manifold. Each orbital arc is a line integral absorbing gravitons. The Coulomb field, on the other hand, is a charge induced diffeomorphism in the gamma ray field. Substantially outside of neutral atoms there is a propensity for positive and negative charges to cancel, though in the near field we have van der Waals forces.

Phonons for the Coulomb interaction are generated inside a charge. The field created, that acts on another charge, may act on the outside of another charge, possibly only 5% of the diameter deep. The fields may also act in the interspace, producing backflush to the charges that generate the fields. Phonons of opposite chirality attract, and of the same chirality repel.

As points meet for the Coulomb force, the acceleration would be periodic, and relates to the vector potential. A Fourier Series can be applied to the vector potential, with the direction of force being the side of the ‘x’ axis where the sine or cosine function has larger lobes. Often a geodesic is called piecewise smooth, due to gravitons being separate, though on a classical scale the motion is smooth.

Two electrons can occupy the same atomic orbital if they have opposite half-integer spin projections. This is the Pauli exclusion principle. In terms of tensor math, “the subspaces are mutually orthogonal and each is a nontrivial irreducible subspace.” ([1] pg. 242)

[1] Boothby, William M., *An Introduction to Differentiable Manifolds and Riemannian Geometry*, Academic Press, 2003

[2] J. Milnor, based on lecture notes by M. Spivak and R. Wells, *Morse Theory*, Princeton University Press, 1969