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).

We start with the following terminology called “a conformal change“.

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}.

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