**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 4 Apr 1999 20:50:14 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 158] An idea for particle modeling"**Previous message:**Matti Pitkanen: "[time 156] Re: [time 150] Re: [time 81] Discreteness and p-adics"**In reply to:**Ben Goertzel: "[time 150] Re: [time 81] Discreteness and p-adics"**Next in thread:**Stephen P. King: "[time 160] Re: [time 157] tangent-cotangent; spaces that is!"

On Sun, 4 Apr 1999, Stephen P. King wrote:

*> Matti,
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*>
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*> Matti Pitkanen wrote:
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*> >
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*> snip
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*>
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*> > There might be something deep in induction of imbedding space
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*> > tangent space octonion structure to spacetime surface [octonion units
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*> > are projected to spacetime and their products which contain also
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*> > part normal to surface are projected to spacetime surface so that one
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*> > obtains tangent space projection C alpha beta gamma of structure constant
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*> > tensor Cklm defined by IkIl = Ckl^mIm ]. But I do not know any idea about
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*> > what deep consequences this might have. Quaternions appear
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*> > in the construction of exact solutions of YM action (instantons): could
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*> > octonions appear in the construction of the absolute minima of Kahler
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*> > action if this construction is possible at all (just a free
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*> > association(;-)?
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*>
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*> Is there a cotangent space here? What relations would exist between the
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*> tangent and cotangent spaces?
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I_k can be regarded as 1-forms and since metric tensor

is present one can map I_k to vector fields I^k by index raising.

I_k is obtained from 'free' octonionic units I_A satisfying standard

octonionic multiplication table by contracting with octobein e^A_k

I_k= e^A_k I_A and this induces structure constant tensor

Ckl^m= e^A_ke^B_ke^Cm C_ABC

Metric is clearly essentially involved and one moves freely between forms

and vectors.

MP

**Next message:**Stephen P. King: "[time 158] An idea for particle modeling"**Previous message:**Matti Pitkanen: "[time 156] Re: [time 150] Re: [time 81] Discreteness and p-adics"**In reply to:**Ben Goertzel: "[time 150] Re: [time 81] Discreteness and p-adics"**Next in thread:**Stephen P. King: "[time 160] Re: [time 157] tangent-cotangent; spaces that is!"

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