I'd like to understand a bit more the following problem.
Suppose a potato shape (say a 3D volume bounded by a 2D surface), and define $n$ the normal vector to its surface, with components $n_i$. Define the object
$$\left\langle \dfrac{n_{i}n_{j}}{1+\alpha_{sr}n_{s}n_{r}}\right\rangle =g_{ij}$$
where repeated indices are summed: for instance in 2D one has $\alpha_{sr}n_{s}n_{r}=\alpha_{11}n_{1}^{2}+\left(\alpha_{12}+\alpha_{21}\right)n_{1}n_{2}+\alpha_{22}n_{2}^{2}$. The notation $\left\langle \cdots\right\rangle $ defines the angular averaging over the surface of the potato: for instance in 2D it will be $\left\langle \cdots\right\rangle =\int_{0}^{2\pi}d\phi\left[\cdots\right]/2\pi$ provided the vector $n$ is $\phi$-dependent. I believe this defines a metric $g_{ij}$. Am I correct on this point ? So the problem should be a problem of differential geometry. Otherwise please make a comment, such that I can modify the question accordingly.
I'd like to know the condition on the matrix $\alpha_{ij}$ which guarantee that $g_{ij}$ is diagonal.
Any help is warm welcome, as well as any remark to improve this question.
A few comments: It seems the question is not trivial. For instance, supposing a circle as the potato, one has $n_{1}=\cos t$ and $n_{2}=\sin t$ only, and the generic $$\alpha=\left(\begin{array}{cc} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{array}\right)$$ and so in this particular example one has
$$g_{ij}=\int_{0}^{2\pi}\dfrac{dt}{2\pi}\left[\dfrac{n_{i}n_{j}}{1+\alpha_{11}\cos^{2}t+\left(\alpha_{12}+\alpha_{21}\right)\cos t\sin t+\alpha_{22}\sin^{2}t}\right]$$
or in matrix form
$$g=\int_{0}^{2\pi}\dfrac{dt}{2\pi}\dfrac{\left(\begin{array}{cc} \cos^{2} t & \cos t\sin t\\ \cos t\sin t & \sin^{2} t \end{array}\right)}{1+\alpha_{11}\cos^{2}t+\left(\alpha_{12}+\alpha_{21}\right)\cos t\sin t+\alpha_{22}\sin^{2}t}$$
which I can not solve for generic $\alpha_{ij}$. It nevertheless seems to me that $\alpha$ needs to be at least antisymmetric to impose a diagonal $g_{ij}$. For instance, suppose $\alpha_{11}=\alpha_{22}$ and $\alpha_{12}=-\alpha_{21}$ one has clearly
$$\left\langle \dfrac{n_{i}n_{j}}{1+\alpha_{11}}\right\rangle =\dfrac{\left\langle n_{i}n_{j}\right\rangle }{1+\alpha_{11}}=\dfrac{\int_{0}^{2\pi}\dfrac{n_{i}n_{j}}{2\pi}dt}{1+\alpha_{11}}=\dfrac{\delta_{ij}}{1+\alpha_{11}}$$
In contrary, one has for instance $$\alpha=\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right)\Rightarrow g=\dfrac{1}{6}\left(\begin{array}{cc} \sqrt{3} & 3-2\sqrt{3}\\ 3-2\sqrt{3} & \sqrt{3} \end{array}\right)$$
Clearly, $\alpha$ being an antisymmetric matrix is a sufficient condition (actually, $\alpha_{22}$ and $\alpha_{11}$ needs to be related as well, since not all values are allowed for these two parameters), but is it necessary ? So I'd like to know if this problem is already known, and eventually solved.