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\begin{document}

Complex vector bundle \( E(M,V_{\rho }) \) with \( V_{\rho }=C^{k} \) structure group is GL(k,C). \( dim_{C}(M)=m \)

total chern class = c(F)=\( det(1+i\frac{F}{2\pi }) \). The determinant is a sum of forms of different
orders: \( =\sum _{j=0}^{min(k,m)}c_{j}(F) \) where \( c_{j} \) is a 2j-form.

G compact means F is anti-hermitian real differential form.

\( F_{ab}=\frac{1}{2}F_{uv}^{\alpha }(\lambda _{\alpha })_{ab}dx^{u}\wedge dx^{v} \)

\begin{itemize}

\item \( c_{0}(F)=1 \)

\item \( c_{1}(F)=itr(\frac{F}{2\pi }) \)

\end{itemize}

\( \frac{iF}{2\pi } \) is hermitian so can be diagonalized by a unitary transformation.

this means \( U^{-1}\frac{iF}{2\pi }U=diag(x_{1},x_{2},...,x_{k}) \) where the \( x_{i} \)'s are 2-forms.

This means \( c(F)=det(diag(1+x_{i},..,1+x_{k}))=\Pi _{i=1}^{k}(1+x_{i}) \)

For the second churn class, you get:

\begin{itemize}

\item \( c_{2}(F)=\sum _{i<j}x_{i}x_{j}=\frac{1}{8\pi ^{2}}[tr(F^{2})-tr^{2}(F)] \)

\end{itemize}

properties:

\begin{itemize}

\item 2 vector bundles, \( E_{1} \) and \( E_{2} \) over same base M, do whitney sum, \( E_{1}\oplus E_{2} \), then
\( c(E_{1}\oplus E_{2})=c(E_{1})c(E_{2}) \)

\item splitting principle 

\end{itemize}

\end{document}
