The reason for my question is here, page 13 in the middle.
Let $P$ be an arbitrary Legendre polynomial, let $x$ be a point on the unit sphere. $S^2$. We identify (and assign an orthonormal basis on) the tangent space $T_x(S^2)$ with $\mathbb{R}^2$ at any $x \neq South$ by fixing an arbitrary orthonormal basis at $North$ and transporting it by the geodesics.
Q: Is it true that
$$\frac{ \partial^2}{\partial e_i^1 \partial e_j^2}[P(\cos(d(._1,._2))](x,x)=0?$$
My progress: Keeping in mind that $\sin(d(x,y))$ disappears when $x=y$, we seem to need to have a look at the (formal) term $P'(\cos d(x,x)) \cos(d(x,x))[(\partial_i^1 d \times \partial_j^2 d)(x,x)]$ which should equal to zero.