Possibly a simple question in differential geometry (maybe not accurate but understandable in mathematical terms): Given an compact surface $ \mathbf {R} $ in $n$ Euclidean space parameterized by $n-1$ variables $ (x_1,x_2,...,x_{n-1}) $ in the following:

$ \mathbf {R} $={ $ X_1,X_2,X_3,...,X_n$ }, ($ X_i=X_i(x_1,x_2,...,x_{n-1}$ ) is the $i$-th Cartesian coordinate)

Then, what is the result of Laplacian operator $∇^2=(1/(\sqrt{g})\partial_{μ}g^{μυ}\sqrt{g} \partial_{υ} $ acting on the $ \mathbf {R} $ as $∇^2 \mathbf {R}$ ? I think that it should be a result that purely depends on the extrinsic curvatures, and also a geometric invariant. Please offer me the result together with a reference which is accessible to a physicist. Thanks.