My Laplace-Beltrami operator isn't transforming correctly under a change of coordinates. What am I doing wrong?
The Laplace-Betrami operator has the following expressing in local coordinates of a Riemannian manifold $(M, g)$,
\begin{align}
\Delta f(x) = g^{ij}(x) \frac{\partial^2 f(x)}{\partial x_i \partial x_j} + g^{jk}\Gamma^i_{jk} \frac{\partial f(x)}{\partial x_i},
\end{align}
where $\Gamma^i_{jk}$ are Christoffel symbols. Written in this way, it is clear that the Laplace-Beltrami operator is a second-order elliptic operator on the manifold.
Many books such as Markov Processes by Dynkin (page 151) or Functional Analysis by Yosida (page 426) state that the coefficient functions of elliptic operators must transform in a prescribed way in order to give a consistent result on the manifold. Namely, consider an elliptic operator of the form (written in local coordinates $x$)
\begin{align}
Af(x) = b^i(x) \frac{\partial f(x)}{\partial x_i} + a^{ij}(x) \frac{\partial^2 f(x)}{\partial x_i\partial x_j}.
\end{align}
Then in another coordinate system $\tilde{x}$ the coefficient functions transform as
\begin{align}
\tilde{b}^i(x) &= b^k(x) \frac{\partial \tilde{x}_i}{\partial x_k} + a^{kl}(x)\frac{\partial^2 \tilde{x}_i}{\partial x_k\partial x_l} \\\\
\tilde{a}^{ij}(x) &= a^{kl}(x) \frac{\partial \tilde{x}_i}{\partial x_k}\frac{\partial \tilde{x}_j}{\partial x_l}.
\end{align}
The Laplace-Beltrami operator should then obey this transformation rule with $a^{ij} =g^{ij}$ and $b^i = g^{jk}\Gamma^i_{jk}$. The fact that $g^{ij}$ obeys the correct transformation rule is apparent; however, I am having a hard time seeing that $g^{jk}\Gamma^i_{jk}$ obeys the correct transformation.
What I want to show is that
\begin{align}
\tilde{g}^{jk}\tilde{\Gamma}^i_{jk} = g^{pq}\Gamma^k_{pq} \frac{\partial \tilde{x}_i}{\partial x_k} + g^{pq}\frac{\partial^2 \tilde{x}_p}{\partial x_k\partial x_q}
\end{align}
I know that the transformation rule for the Christoffel symbols is as follows:
\begin{align}
\tilde{\Gamma}^i_{jk} &= \frac{\partial x_p}{\partial\tilde{x_j}}\frac{\partial x_q}{\partial\tilde{x}_k} \Gamma_{pq}^r \frac{\tilde{x}_i}{\partial x_r} + \frac{\partial^2 x_r}{\partial \tilde{x}_j\partial \tilde{x}_k} \frac{\partial \tilde{x}_i}{\partial x_r} \\
&= \frac{\partial x_p}{\partial\tilde{x_j}}\frac{\partial x_q}{\partial\tilde{x}_k} \Gamma_{pq}^r \frac{\tilde{x}_i}{\partial x_r} – \frac{\partial x_p}{\partial \tilde{x}_j} \frac{\partial^2 \tilde{x}_i}{\partial x_p\partial x_q} \frac{\partial x_q}{\partial \tilde{x}_k}
\end{align}
If I multiply both sides by $\tilde{g}^{jk}$ and use the transformation law of the inverse metric in coordinates, I obtain,
\begin{align}
\tilde{g}^{jk} \tilde{\Gamma}^i_{jk} &= \tilde{g}^{jk} \frac{\partial x_p}{\partial\tilde{x_j}}\frac{\partial x_q}{\partial\tilde{x}_k} \Gamma_{pq}^r \frac{\tilde{x}_i}{\partial x_r} – \tilde{g}^{jk}\frac{\partial x_p}{\partial \tilde{x}_j} \frac{\partial^2 \tilde{x}_i}{\partial x_p\partial x_q} \frac{\partial x_q}{\partial \tilde{x}_k} \\
&= g^{pq} \Gamma_{pq}^r \frac{\tilde{x}_i}{\partial x_r} – g^{pq} \frac{\partial^2 \tilde{x}_i}{\partial x_p\partial x_q}.
\end{align}
This is very nearly what I wanted to show, but differs from the expected result by a negative sign in the second term. What have I done wrong?
Best Answer
Even though this is an old question, this could hopefully help someone in the future.
If we compute the Laplacian from its coordinate-free definition $\Delta f = \mathrm{tr}~\nabla (\mathrm{grad}~u)$ and we consider the definition of the Christoffel symbols, we arrive at $$ \Delta u = (\partial_i\partial_ju)g^{ij} + (\partial_iu)\partial_j(g^{ij}) + (\partial_iu)g^{ij}\Gamma_{kj}^k. $$
This is still not quite the expression you presented us, so we have to use the old trick $g^{ij}g_{jk}=\delta^i_k$ and consider the expression of the Christoffel symbols in terms of the coefficients of the metric $$ \Gamma_{jk}^i = \frac{g^{im}}{2} \left[ \partial_j(g_{mk}) + \partial_k(g_{mj}) - \partial_m(g_{jk}) \right] $$ in order to re-write such expression as $$ \Delta u = (\partial_i\partial_ju)g^{ij} - (\partial_iu)g^{jk}\Gamma_{jk}^i. $$
As this matches the expression of the Laplace-Beltrami Operator given, for example, in List of formulas in Riemannian geometry - Wikipedia, then I believe your mistake was the local expression that you considered the Laplace-Beltrami Operator to have, and not your computations.