I am seeing this equation, but I am not following how it is possible.
It mentions the Laplace-Beltrami operator (here we assume it is a flat manifold) and at the last part using the Einstein Summation Convention.
x is a vector and $\mu$ is the mean of it. g being a covariance matrix.
$\Delta_g f(x):=\sum_{i,j}E((x_i-\mu_i)(x_j-\mu_j))\partial_i\partial_jf=\sum_{i,j}g^{i,j}\partial_j\partial_jf=:\partial_j\partial^jf$
I am especially confused how the $\partial_i\partial_j$ in front of f can change to $\partial_i\partial_i$.
I am also confused how the g can be removed in the end.
I am sorry for such incomplete problem, but I am not understanding what is going on, and thinking someone who knows might be able to hint me what to look into.