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

*Mon, 5 Apr 1999 07:56:33 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 170] Re: [time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Previous message:**Hitoshi Kitada: "[time 168] Re: [time 164] Question"**In reply to:**Ben Goertzel: "[time 164] Question"**Next in thread:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"

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

*> Matti,
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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 am not familiar with the meaning, e.g. I think visually, of what your
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*> reply meams. :(
*

Yes there is: quaternion units I_k could be regarded as local sections of

cotangent bundle/one-forms or covariant vector fields in terminology of

physicist. In any case, when one has Riemann metric one can move between

contangent and tangent spaces freely: they are one and same thing, one

could say. I visualize I^k as unit vectors in directions of various

coordinate axis: the relationship between basis partial_k and dx^k for

cotangent and tangent spaces reads as partial_k(dx^l)= delta_k^l and

translates to Re(I_kI^l)=delta_k^l: note that there is independence on

metric signature.

MP

*>
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*> Stephen
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*>
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*> Matti Pitkanen wrote:
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*> >
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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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*>
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**Next message:**Matti Pitkanen: "[time 170] Re: [time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Previous message:**Hitoshi Kitada: "[time 168] Re: [time 164] Question"**In reply to:**Ben Goertzel: "[time 164] Question"**Next in thread:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"

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