A reformulation of diff. geo. making reference to tangent spaces. This is essential if there are spinors (fermi fields), and nice even if there are no fermions.

Define tex2html_wrap132 as spanned by tex2html_wrap133 .

Can form new basis, tex2html_wrap134 Demand tex2html_wrap135

tex2html_wrap136 where tex2html_wrap137

Can rewrite any tex2html_wrap138 where tex2html_wrap139 and tex2html_wrap140 .

This can be extended to the cotangent space, tex2html_wrap141 easily. Find tex2html_wrap142 s.t. tex2html_wrap143 . Then tex2html_wrap144 .

In local coordinates: tex2html_wrap145

Example:

tex2html_wrap146 , tex2html_wrap147

let tex2html_wrap148 , tex2html_wrap149 . Then tex2html_wrap150 tex2html_wrap151

vielbeins are redundant. A rotation of the basis has the same metric.


source
psfile jl@crush.caltech.edu index