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\begin{document}

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 \( T_{p}M \)as spanned by \( \partial _{u} \).

Can form new basis, \( e_{a}=e_{a}^{u}\partial _{u} \) Demand \( g(e_{a},e_{b})=\delta _{ab} \)

\( g_{uv}=\delta _{ab}e^{a}_{u}e^{b}_{v} \) where \( e^{a}_{u}e^{u}_{b}=\delta _{ab} \)

Can rewrite any \( V=V^{u}\partial _{u}=V^{a}e_{a} \) where \( V^{a}=e^{a}_{u}V^{u} \)and \( V^{u}=e^{u}_{a}V^{a} \).

This can be extended to the cotangent space, \( T^{*}_{p}M \)easily. Find \( \theta  \)s.t. \( <e_{a},\theta ^{b}>=\delta _{a}^{b} \).
Then \( g^{-1}(\theta ^{a},\theta ^{b})=\delta ^{ab} \).

In local coordinates: \( \theta ^{a}=e^{a}_{u}dx^{u} \)

Example:

\( S^{2} \), \( ds^{2}=d\theta ^{2}+sin^{2}(\theta )d\phi ^{2} \)

let \( t^{1}=d\theta  \), \( t^{2}=sin(\theta )d\phi  \). Then \( e^{a}_{u}= \)\( \begin{array}{cc}
1 & 0\\
0 & sin(\theta )
\end{array}
 \)

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

\end{document}