dg.differential geometry - Looking for a reference for the laplacian operator

So I wrote up this small derivation drawing insights from the answer by Deane Yang and Steve Huntsman,

With respect to the Riemann-Christoffel connection on a riemannian manifold the laplacian on that manifold will have the form

$$nabla ^2 phi = frac{1}{sqrt{g}} partial _{mu} left [ sqrt{g} g^{mu nu} partial _{nu} phi right ]$$

where $g$ is the the determinant of the metric on the manifold and $phi$ is some smooth scalar function on the manifold.

On can write the line element on $S^n subset mathbb{R}^{n+1}$ as,

$dOmega _n ^2 = dtheta _1 ^2 + sin^2 theta_1 dtheta _2 ^2 + sin^2 theta _1 sin^2 theta_2 dtheta _3 ^3 +...+sin^2theta _1 sin^2 theta_2...sin^2 theta_{n-2} sin^2 theta_{n-1} dtheta _n ^2$

Then the line element on $mathbb{R}^{n+1}$ in polar coordinates can be written as,

$$ds^2 = dr^2 + r^2 dOmega _n ^2$$

and
$g_{mathbb{R}^{n+1}} = r^{2n}g_{_{S^n}}$
where
$g_{_{S^n}} = (sin^2 theta _1)^{n-1}(sin^2 theta_2)^{n-2}...(sin^2 theta _{n-2})^2(sin^2 theta_{n-1})^1$

Therefore since the metric is diagonal $nabla _{mathbb{R}^{n+1}} ^2 phi = frac{1}{r^n sqrt{g_{_{S^n}} }} partial _{mu} left [r^n sqrt{g_{_{S^n}} } g^{mu mu}_{mathbb{R}^{n+1}} partial _{mu} phi right ]$

$=frac{1}{r^n sqrt{g_{_{S^n}} }} partial _{r} left [r^n sqrt{g_{_{S^n}} } partial _{r} phi right ] + frac{1}{r^n sqrt{g_{_{S^n}} }} partial _{theta _i} left [r^n sqrt{g_{_{S^n}} } g^{theta _i theta _i}_{mathbb{R}^{n+1}} partial _{theta _i} phi right ]$

$=frac{1}{r^n}partial _r (r^n partial_r phi) + frac{1}{sqrt{g_{_{S^n}} }} partial_{theta _i} left[ sqrt{g_{_{S^n}} } frac{g^{theta _i theta _i}_{S^n}} {r^2} partial _{theta _i} phi right]$

$=frac{1}{r^n}partial _r (r^n partial_r phi) + frac{nabla _{S^n}^2 phi}{r^2}$

Therefore after doing the differentiation we have the final result,

$$nabla _{mathbb{R}^{n+1}}^2 phi = frac{n}{r}partial _r phi + partial _r ^2 phi + frac{nabla _{S^n} ^2 phi}{r^2}$$

And I don't see an neat way of writing the Laplacian on $S^n$ !

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