**Stephen P. King** (*stephenk1@home.com*)

*Sun, 04 Apr 1999 14:13:46 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Previous message:**Ben Goertzel: "[time 159] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"**In reply to:**Matti Pitkanen: "[time 154] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"**Next in thread:**Matti Pitkanen: "[time 169] Re: [time 160] Re: [time 157] tangent-cotangent; spaces that is!"

Matti,

Is there a cotangent space here? What relations would exist between the

tangent and cotangent spaces?

I am not familiar with the meaning, e.g. I think visually, of what your

reply meams. :(

Stephen

Matti Pitkanen wrote:

*>
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*> On Sun, 4 Apr 1999, Stephen P. King wrote:
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*>
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*> > 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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*>
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*> I_k can be regarded as 1-forms and since metric tensor
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*> is present one can map I_k to vector fields I^k by index raising.
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*>
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*> I_k is obtained from 'free' octonionic units I_A satisfying standard
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*> octonionic multiplication table by contracting with octobein e^A_k
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*>
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*> I_k= e^A_k I_A and this induces structure constant tensor
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*>
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*> Ckl^m= e^A_ke^B_ke^Cm C_ABC
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*>
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*> Metric is clearly essentially involved and one moves freely between forms
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*> and vectors.
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*>
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*> MP
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**Next message:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Previous message:**Ben Goertzel: "[time 159] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"**In reply to:**Matti Pitkanen: "[time 154] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"**Next in thread:**Matti Pitkanen: "[time 169] Re: [time 160] Re: [time 157] tangent-cotangent; spaces that is!"

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