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.


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.

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  • $\begingroup$ Thank a lot for the answer. $D_X\omega=0$ is a reasonable assumption if $D$ is the standard covariant derivative on $\mathrm R^3$, and $\omega$ is the determinant. Thanks again. $\endgroup$ – Fly by Night Apr 11 '16 at 19:38

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