Hessian of the Stereographic projection

Question

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in $\mathbb{R}^{n + 1}$). I am trying to see if there is any relation between the Hessian of the stereographic projection and the Christoffel symbols of $S^n$ (both calculated in the polar coordinates); in particular, if they are equal up to a constant. If this is true, is there a slick method of seeing this quickly rather than computing everything explicitly? Thanks for any help.