I'm reading Nomizu & Sasaki's "*Affine Differential Geometry: Geometry of Affine Immersions*" and I'm having some trouble with Proposition 1.4.


I have an immersed surface in $M \hookrightarrow \mathbb R^{3}$. Assume that we have a volume form, say $\omega$ on $\mathbb R^{3}$. This could, for example, be the determinant. We also have a covariant derivative on $\mathbb R^{3}$, say $D$.


Let $Y$ and $Z$ be smooth, linearly independent vector fields on $M$, and let $\xi$ be a smooth transverse vector field on $M$, i.e. $T\_pM \oplus \mathrm{span}(\xi) = T\_p\mathbb R^3$ for all $p \in M$. We have the two equations
\begin{eqnarray\*}
D\_YZ &=& \nabla\_YZ + h(Y,Z)\xi \\ \\
D\_Y\xi &=& -SY + \tau(Y)\xi
\end{eqnarray\*}
We can define a volume element on $M$, say $\theta$, as $\theta(Y,Z) := \omega(Y,Z,\xi)$.


**Proposition 1.4:** $\nabla\_X\theta = \tau(X)\theta$ for all $X \in T\_pM$.


**My attempt to follow the proof:**


$$X[\omega(Y,Z,\xi)] = [D\_X\omega](Y,Z,\xi)+\omega(D\_XY,Z,\xi)+\omega(Y,D\_XZ,\xi)+\omega(Y,Z,D\_X\xi)$$
where $X[\omega(Y,Z,\xi)]$ is the directional derivative of the function $\omega(Y,Z,\xi)$. 


First, notice that $X[\omega(Y,Z,\xi)]=X[\theta(Y,Z)]$. Second 
\begin{eqnarray\*}
\omega(D\_XY,Z,\xi) &=& \omega(\nabla\_XY+h(X,Y)\xi,Z,\xi) \\ \\
&=& \omega(\nabla\_XY,Z,\xi) \\ \\
&=& \theta(\nabla\_XY,Z)
\end{eqnarray\*}
Similarly $\omega(Y,D\_XZ,\xi) = \theta(Y,\nabla\_XZ)$. Finally we have
\begin{eqnarray\*}
\omega(Y,Z,D\_X\xi) &=& \omega(Y,Z,-SX+\tau(X)\xi) \\ \\
&=& \omega(Y,Z,\tau(X)\xi) \\ \\
&=& \tau(X)\omega(Y,Z,\xi) \\ \\
&=& \tau(X)\theta
\end{eqnarray\*}
Substituting all of these simplifications into the original equations gives
$$X[\theta(Y,Z)] = [D\_X\omega](Y,Z,\xi)+\theta(\nabla\_XY,Z)+\theta(Y,\nabla\_XZ)+\tau(X)\theta(Y,Z)$$


**Where I get lost:**


The authors then suddenly jump to this:
\begin{eqnarray\*}
[\nabla\_X\theta](Y,Z) &=& X[\theta(Y,Z)] - \theta(\nabla\_XY,Z)-\theta(Y,\nabla\_XZ) \\ \\
&=& \tau(X)\theta(Y,Z)
\end{eqnarray\*}


I think they have done something with $[D\_X\omega](Y,Z,\xi)$ but I don't know what. 



Proof about affine connections
The formulas all look correct. With the definition of $\nabla\_X\theta$, one gets
\begin{align\*}
 [\nabla\_X\theta](Y,Z)&=X[\theta(Y,Z)]-\theta(\nabla\_XY,Z)-\theta(Y,\nabla\_XZ)\\
 &=X[\omega(Y,Z,\xi)]-\omega(D\_XY,Z,\xi)-\omega(Y,D\_XZ,\xi)-\omega(Y,Z,D\_X\xi)+\tau(X)\theta(Y,Z)\\
&=[D\_X\omega](Y,Z,\xi)+\tau(X)\theta(Y,Z)\;.
\end{align\*}
So the missing thing is the assumption that $D\_X\omega=0$, that is,
the covariant derivative is volume preserving.


