Stephen P. King (firstname.lastname@example.org)
Sun, 04 Apr 1999 14:13:46 -0400
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. :(
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.
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