Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 5 Apr 1999 07:56:33 +0300 (EET DST)
On Sun, 4 Apr 1999, Stephen P. King wrote:
> 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. :(
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
>
> Stephen
>
> Matti Pitkanen wrote:
> >
> > On Sun, 4 Apr 1999, Stephen P. King wrote:
> >
> > > Matti,
> > >
> > > Matti Pitkanen wrote:
> > > >
> > > snip
> > >
> > > > There might be something deep in induction of imbedding space
> > > > tangent space octonion structure to spacetime surface [octonion units
> > > > are projected to spacetime and their products which contain also
> > > > part normal to surface are projected to spacetime surface so that one
> > > > obtains tangent space projection C alpha beta gamma of structure constant
> > > > tensor Cklm defined by IkIl = Ckl^mIm ]. But I do not know any idea about
> > > > what deep consequences this might have. Quaternions appear
> > > > in the construction of exact solutions of YM action (instantons): could
> > > > octonions appear in the construction of the absolute minima of Kahler
> > > > action if this construction is possible at all (just a free
> > > > association(;-)?
> > >
> > > Is there a cotangent space here? What relations would exist between the
> > > tangent and cotangent spaces?
> >
> > 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
>
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