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\begin{document}

The Hodge dual allows the definition of an inner product for \( w,n\in \Omega ^{r} \), define
\( (w,n)=\int _{M}w\wedge *n \). \( w\wedge *n=\frac{1}{r!}w_{u_{1}..u_{r}}n^{u_{1}...u_{r}}\Omega _{M} \). satisfies

\begin{itemize}

\item \( (w,n)=(n,w) \)

\item \( (w,w)>0 \) \( \forall w\neq 0 \)on positive definite metrics.

\end{itemize}

Now can define \( w\in \Omega ^{r},n\in \Omega ^{r-1} \), require \( (w,dn)=(d^{\dagger }w,n) \).

Given (M,g) with \( \partial M=0 \), \( (d^{\dagger }w,n)= \) \( (n,d^{\dagger }w)= \)\( \int _{M}n\wedge d^{\dagger }w \)

\( =(w,dn)=(dn,w)=\int _{M}dn\wedge *w \) \( =\int _{M}[d(n\wedge *w)-(-1)^{r-1}n\wedge d(*w)] \) \( =\int _{M}-(-1)^{r-1}n\wedge d(*w) \)

so \( **d^{\dagger }w=(-1)^{r}*d(*w) \) \( =(-1)^{(r-1)(m-r-1)}d^{\dagger }w \). also \( d^{\dagger }w=(-1)^{m(r+1)+1}*d*w \)in the euclidean case.

\( *:r\rightarrow m-r \), \( d:m-r\rightarrow m-r+1 \), \( *:m-r+1\rightarrow r-1 \).

properties of \( d^{\dagger } \)

\begin{itemize}

\item \( (d^{\dagger })^{\dagger }=d \)

\item \( (d^{\dagger })^{2}=0 \) so \( d^{\dagger } \) is nilpotent.

\item if \( d^{\dagger }w=0 \)w is co-closed

\item \( w=d^{\dagger }n \) then w is co-exact

\end{itemize}

\end{document}
