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## let’s push for understanding what’s going on with the CODAZZI FORMULA

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:07 PM (24 minutes ago)

 to aimee
So the highlighted formula in the snapshot is taking the covariant derivative of the second fundamental form B(X,Y).  The Codazzi formula says our context of a hypersurface M of S4(1/h) and B(A,B,C) := g(B(A,B),C)

D_X B(A,B,C) – D_A B(X,B,C) = RiemannCurvatureTensorOfSubmanifold(X,A,B,C)
Hmm, this will require some more thought.
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Preview attachment Screenshot 2015-06-26 21.57.49.png

Screenshot 2015-06-26 21.57.49.png

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:08 PM (22 minutes ago)

 to aimee
This has to be resolved probably by tomorrow and another day gone.  I need rest of the mind.  Nice progress though.

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:18 PM (12 minutes ago)

 to David, bcc: aimee
Actually we need not do this on the level of functions at all as spinors are better and for this Hijazi already has an explicit formula for the potential in terms of the shape operator/second fundamental form.

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:25 PM (6 minutes ago)

 to David, bcc: aimee
So Dirac operator on a three-dimensional hypersurface is adjusted from that of the ambient sphere by a term like

gamma(AN)gamma(N)
which is then the potential term and this is the spinorial potential that we claim to be the UNIVERSAL POTENTIAL because all matter and energy in the material universe arise geometrically in an S4 universe following the simple fundamental equation of S4 physics which was the Dirac^2 wave equation.  This would be a pretty interesting result if checked experimentally or by furthering this S4 theory of physics.  This provides immediate correction to the Dirac equation by showing the exact manner in which nonlinearity appears GEOMETRICALLY in a deterministic universe.

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:31 PM (0 minutes ago)

 to David
Oops the exact formula is (8) in the snapshot the above is the correction for covariant derivative not for the Dirac operator.

This is the Hijazi paper which is a nice math paper but the explicit potential term for Dirac operator in terms of Clifford multiplication was known to Witten, so this is known.  The interesting and new claim about these potentials are that they are the only ones that are actually physical potentials.  Therefore whether one accepts this mathematical physics theory as valid or not, we have a deterministic ‘grand unification’ since both electromagnetism and gravity are handled geometrically and simply and the Dirac equation for flat 3-space is not difficult to see would appear in linear approximations of the fundamental Dirac wave equation on S4 restricted to a physical universe hypersurface.
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