# Ngô Quốc Anh

## May 8, 2011

### Commutator of Delta and Gradient on functions

Filed under: Riemannian geometry — Ngô Quốc Anh @ 0:05

Let us prove following interesting identity between $\Delta$ and $\nabla$

$\Delta \nabla_i f=\nabla_i \Delta f+{\rm Ric}_{ij}\nabla_j f$

for any function $f$.

For any function $f$, using the formula

$\Delta = g^{ij}\nabla_i \nabla_j$

we obtain

$\Delta {\nabla _i}f = {g^{mn}}{\nabla _m}{\nabla _n}({\nabla _i}f).$

Since $f$ is a funtion, we know that

${\nabla _n}({\nabla _i}f) = {\nabla _i}({\nabla _n}f)$

then we obtain

$\Delta {\nabla _i}f = {g^{mn}}{\nabla _m}({\nabla _i}({\nabla _n}f)).$

Now $\nabla_n f$ is a vector, we need to use the Riemmanian curvature tensor, which measures the difference between $\nabla_m \nabla_i$ and $\nabla_i\nabla_m$.

${\nabla _m}({\nabla _i}({\nabla _n}f)) = {\nabla _i}({\nabla _m}({\nabla _n}f)) + {\rm Rm}({e^m},{e^i}){\nabla _n}f.$

Notice that there was an extra term involving Lie bracket. Fortunately, that term vanishes. Thus

$\Delta {\nabla _i}f = {g^{mn}}{\nabla _i}({\nabla _m}({\nabla _n}f)) + {g^{mn}}{\rm Rm}({e^m},{e^i}){\nabla _n}f = {\nabla _i}\Delta f + {\rm Ric}_{ik}{\nabla _k}f.$

The proof follows.

1. I am wondering that why $\nabla_n f$ is a vector?

Comment by lucky — February 5, 2013 @ 20:14

• Hi lucky, thanks for your comment, to be exact, I should say that $\nabla f$ is a vector field instead of $\nabla_n f$. However, as components, we often use $\nabla_nf$ to denote the vector field $\nabla f$ since

$\displaystyle\nabla f = \underbrace {{g^{nj}}\frac{{\partial f}}{{\partial {x^j}}}}_{{\nabla _n}f}\frac{\partial }{{\partial {x^n}}}.$

Comment by Ngô Quốc Anh — February 5, 2013 @ 21:08

2. Hi Ngo!, To your knowledge,is there an analog relation between the Laplacian and the bi-Laplacian?
F

Comment by Fab — September 28, 2013 @ 23:13

• Dear Fab,

I have not seen anything like that once. Fortunately, we can derive a similar formula by taking another $\Delta$ to get

$\displaystyle \Delta^2 \nabla_i f = \nabla_i (\Delta^2 f) + \text{Ric}_{ij} \nabla_j f + \text{Ric}_{ij} \nabla_j (\Delta f).$

Comment by Ngô Quốc Anh — September 28, 2013 @ 23:19

3. Just a curiosity, how can I write down the bi-Laplacian on a manifold?

Comment by Fab — September 29, 2013 @ 21:28

• Dear Fab,

I don’t think we have a very precise formula for $\Delta^2$ in local coordinates so far as its calculation is quite complicated and possibly useless. Instead, you may use

$\Delta^2 u = g^{ij}\nabla_i \nabla_j (g^{pq}\nabla_p \nabla_q u) = \nabla_i\nabla_i (\nabla_j \nabla_j u)$

to deal with.

Comment by Ngô Quốc Anh — September 29, 2013 @ 22:30