# Ngô Quốc Anh

## April 24, 2010

### Conformal invariant operators: Laplacian operators

Filed under: Riemannian geometry — Ngô Quốc Anh @ 9:23

Let $M$ be an $n$-dimensional differentiable connected Riemannian manifold with metric tensor $g$. In order to distinguish $g$ from other metrics on $M$, we shall denote the manifold $M$ with $g$ by $(M, g)$.

Definition. If $g'$ is another metric on $M$, and there is a function $\omega$ on $M$ such that $g' = e^{2\omega}g$, then $g$ and $g'$ are said to be conformally related or conformal to each other, and such a change of metric $g \to g'$ is called a conformal change of Riemannian metric.

We are now in a position to define the so-called “conformal invariant operators“.

Definition. We call a metrically de fined operator $A$ conformally covariant of bi-degree $(a,b)$ if under the conformal change of metric $g_\omega = e^{2\omega}g$, the pair of  corresponding operators $A_{g_\omega}$ and $A$ are related by

$\displaystyle {A_{{g_\omega }}}(\varphi ) = {e^{ - b\omega }}A({e^{a\omega }}\varphi ), \quad \forall \varphi \in {C^0}(M)$.

Example 1. The Laplace-Beltrami operator is conformal invariant.

Proof. Let us recall from this topic that on $(M,g)$, the Laplace-Beltrami operator is defined to be

$\displaystyle \Delta_g ={\rm div}({\rm grad \left(\cdot\right)})$

and in local coordinates it is given as follows

$\displaystyle\Delta_g = \frac{1}{{\sqrt {\det \left( g\right)} }}\frac{\partial }{{\partial {x^j}}}\left( {\sqrt {\det \left( g\right)} {g^{ij}}\frac{{\partial}}{{\partial {x^i}}}} \right)$.

By a change of metric $g_\omega = e^{2\omega}g$ we need to calculate $\Delta_{g_\omega}$ and then compare with $\Delta_g$.

Indeed, in local coordinates

$\displaystyle {g_\omega } = {e^{2\omega }}{g_{ij}}d{x^i} \otimes d{x^j}$

which implies

$\displaystyle \det ({g_\omega }) = \det ({e^{2\omega }}{g_{ij}}) = {e^{2\omega }}\det (g)$.

Besides,

$\displaystyle {({g_\omega })^{ij}} = {({g_\omega })^{ - 1}} = {({e^{2\omega }}{g_{ij}})^{ - 1}} = \frac{1}{{{e^{2w}}}}{g^{ij}}=e^{-2\omega}g^{ij}$.