I am working with an equation of the form $-\Delta_{\mathbb{S}^n} u =f(u),$ where $u:\mathbb{S}^n\to \mathbb{R}$ and $n\geq 3.$ To understand this equation better, I would like to consider the case when the function $u(\omega)$ where $\omega=(\omega_1,\omega_2,\cdots,\omega_{n+1})$ depends only on one variable, say $\omega_{n+1}.$ Note here we are viewing $\mathbb{S}^n$ has a submanifold of $\mathbb{R}^{n+1}.$
If this was the usual Laplacian on $\mathbb{R}^n$ then $-\Delta_{\mathbb{R}^n}u(x_{n})=-\partial_{x_n x_n}u.$ However from what I understand from Wikipedia is that
$${\displaystyle \Delta _{S^{n}}f(\xi ,\phi )=(\sin \phi )^{1-n}{\frac {\partial }{\partial \phi }}\left((\sin \phi )^{n-1}{\frac {\partial f}{\partial \phi }}\right)+(\sin \phi )^{-2}\Delta _{\xi }f}$$
where $(\phi,\zeta)$ are normal co-ordinates. So I am guessing that if $u=u(\phi)$ then $$\Delta_{\mathbb{S}^n}u(\phi)=(\sin \phi )^{1-n}{\frac {\partial }{\partial \phi }}\left((\sin \phi )^{n-1}{\frac {\partial u}{\partial \phi }}\right)=u''+\frac{(n-1)}{\sin(\phi)}u'.$$
Is this computation correct?