conformal changes « Ngô Quốc Anh

May 16, 2010

Conformal Changes of Riemannian Metrics

Filed under: Riemannian geometry — Tags: conformal changes — Ngô Quốc Anh @ 5:54

I guess I will use the relation between curvature tensors of metrics lying in a conformal class frequently so I decide to post something related to this stuff which may be helpful and we can use later. Actually, I have used it when we proved conformal Laplacian operator is invariant. Let us briefly recall some terminologies

Definition (conformal). Two pseudo-Riemannian metrics $g$ and $\widetilde g$ on a manifold $M$ are said to be

(pointwise) conformal if there exists a $C^\infty$ function $f$ on $M$ such that
$\displaystyle \widetilde g=e^{2f}g$ ;

conformally equivalent if there exists a diffeomorphism $\alpha$ of $M$ such that $\alpha^* \widetilde g$ and $g$ are pointwise conformal.

Note that, if $g$ and $\widetilde g$ are conformally equivalent, then $\alpha$ is an isometry from $e^{2f}g$ onto $\widetilde g$ . So we will only study below the case $\widetilde g = e^{2f}g$ . Our aim is to compare Riemann curvature, Scalar curvature, Ricci curvature,… of $g$ and $\widetilde g$ .

Definition (the Kulkarni–Nomizu product). This product $\odot$ is defined for two $(0,2)$ -tensors and gives as a result a $(0,4)$ -tensor. Precisely,

$\displaystyle \alpha \odot \beta (X_1,X_2,X_3,X_4)=\alpha (X_1,X_3)\beta (X_2,X_4)+\alpha (X_2,X_4)\beta (X_1,X_3)-\alpha (X_1,X_4)\beta (X_2,X_3)-\alpha (X_2,X_3)\beta (X_1,X_4)$

or

$\displaystyle {(\alpha \odot \beta )_{ijkl}} = {\alpha _{il}}{\beta _{jk}} + {\alpha _{jk}}{\beta _{il}} - {\alpha _{ik}}{\beta _{jl}} - {\alpha _{jl}}{\beta _{ik}}$ .

Levi-Civita connection. On $(M,g)$ , the Levi-Civita connection $\nabla$ is an affine connection which is torsion free

$\displaystyle \nabla_XY+\nabla_YX=[X,Y]$

and satisfies the rule

$\displaystyle X(g(Y,Z))=g(\nabla_X Y,Z) + g(Y, \nabla_X Z)$

for any vector fields $X,Y,Z$ . We now have

$\displaystyle {\widetilde\nabla _X}Y = {\nabla _X}Y + X(f)Y + Y(f)X - g(X,Y) {\rm grad}f$ .

Weyl tensor. This tensor is defined to be

$\displaystyle W = R - \frac{1}{{n - 2}}\left( {{\rm Ric} - \frac{{\rm Scal}}{n}g} \right) \odot g - \frac{{\rm Scal}}{{2n(n - 1)}}g \odot g$ .

Thus we have the rule

$\displaystyle\widetilde W =W$ .

Ricci tensor. This is a $(2,0)$ -tensor defined by

$\displaystyle {\rm Ric}(X,Y) = {\rm Trace}( x \to R(x, X)Y)$ .

In local coordinates, it has the form

$\displaystyle {\rm Ric} = R_{ij} dx^i \otimes dx^j$ .

So we have the following rule

$\displaystyle\widetilde{\rm Ric} = {\rm Ric} - (n - 2)({\rm Hess} f -{\rm grad}f \otimes {\rm grad}f) + (\Delta f - (n - 2)|{\rm grad}f|^2)g$ .

Traceless Ricci tensor. This tensor is defined by

$\displaystyle\displaystyle {Z_{ij}} = {R_{ij}} - \frac{1}{n}{\rm Scal}{g_{ij}}$ .

A simple calculation shows that its trace, $g^{ij}Z_{ij}$ , equals zero. So

$\displaystyle\widetilde Z = Z - (n - 2)\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f} \right) - \frac{{n - 2}}{n}\left( {\Delta f + |{\rm grad}f|^2} \right)g$ .

Scalar curvature. This $(2,0)$ tensor is defined to be the trace of Ricci tensor, that is

$\displaystyle {\rm Scal} = {\rm Trace}( {\rm Ric}) = g^{jk}{\rm Ric}_{jk}$ .

$\displaystyle\widetilde{\rm Scal} = {e^{ - 2f}}\left[ {{\rm Scal} + 2(n - 1)\Delta f - (n - 2)(n - 1)|{\rm grad} f{|^2}} \right]$ .

In practice, this conformal change is not useful, we usually use the following conformal change

$\displaystyle \widetilde g=f^\frac{4}{n-2}g$ .

With this, we simply have

$\displaystyle - \Delta f + \frac{{n - 2}}{4(n - 1)}{\rm Scal}f = \frac{{n - 2}}{4(n - 1)}\widetilde{{\rm Scal}}{f^{\frac{{n + 2}}{{n - 2}}}}$

or

$\displaystyle \widetilde{{\rm Scal}} = {f^{ - \frac{{n + 2}}{{n - 2}}}}\left[ { - \frac{4(n - 1)}{n - 2}\Delta f + {\rm Scal}f} \right]$ .

Riemann curvature tensor. This $(1,3)$ tensor is defined to be

$\displaystyle R(X,Y)Z = \nabla_X \nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z$ .

In local coordinates, we get

$\displaystyle R = R_{ikl}^j\dfrac{\partial }{{\partial {x^j}}} \otimes d{x^i} \otimes d{x^k} \otimes d{x^l}$ .

$\displaystyle \widetilde R = {e^{2f}}\left[ {R - g\odot\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f + \frac{1}{2}|{\rm grad}f{|^2}g} \right)} \right]$ .

Volume element. This, $d{\rm vol}_g$ , is the unique density such that, for any orthonormal basis $(X_i)$ of $T_XM$ ,

$\displaystyle d{\rm vol}_g(X_1,...,X_n)=1$ .

In local coordinates,

$\displaystyle d{\rm vol}_g = \sqrt{|g|} dx^1\wedge \dots \wedge dx^n$ .

$\displaystyle d{\rm vol}_{\widetilde g} = e^{nf}d{\rm vol}_g$ .

Hodge operator on $p$ -forms (if $M$ is oriented). The Hodge star operator on an oriented inner product space $V$ is a linear operator on the exterior algebra of $V$ , interchanging the subspaces of $k$ -vectors and $n-k$ -vectors where $n = \dim V$ , for $0 \leqslant k \leqslant n$ . It has the following property, which defines it completely: given an oriented orthonormal basis $e_1,e_2,\dots,e_n$ we have

$\displaystyle *(e_{i_1} \wedge e_{i_2}\wedge \cdots \wedge e_{i_k})= e_{i_{k+1}} \wedge e_{i_{k+2}} \wedge \cdots \wedge e_{i_n}$ .

One can repeat the construction above for each cotangent space of an $n$ -dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual $n-k$ -form, of a $k$ -form. The Hodge star then induces an $L^2$ -norm inner product on the differential forms on the manifold. One writes

$\displaystyle (\eta,\zeta)=\int_M \eta\wedge *\zeta$

for the inner product of sections $\eta$ and $\zeta$ of $\Lambda^k(M)$ . (The set of sections is frequently denoted as $\Omega^k(M) = \Gamma(\Lambda^k(M)$ ). Elements of $\Omega^k(M)$ are called exterior $k$ -forms). For example, for a positively oriented orthogonal cofram $\{\omega^i\}_1^n$ , one has

$\displaystyle *(\omega^1 \wedge \cdots \wedge \omega^p)=\omega^{p+1}\wedge \cdots \wedge \omega^n$ .

$\displaystyle {*_{\widetilde g}} = {e^{(n - 2p)f}}{*_g}$ .

Codifferential on $p$ -forms. This notion $\delta$ is usually defined through the exterior derivative

$\displaystyle d : \Omega^p(M) \to \Omega^{p+1}(M)$

by the following rule (also called the formal adjoint of exterior derivative)

$\displaystyle\langle \eta,\delta \zeta\rangle = \langle d\eta,\zeta\rangle$ ,

i.e.

$\displaystyle \delta : \Omega^p(M) \to \Omega^{p-1}(M)$ .

In other words, for a $p$ -form $\beta$ ,

$\displaystyle \delta \beta = {( - 1)^{np + n + 1}}*d*\beta$ .

$\displaystyle\widetilde\delta \beta = {e^{ - 2f}}\left[ {\delta \beta - (n - 2p){\iota_{{\rm grad}f}}\beta } \right]$

where $\iota$ denotes the interior product (the contraction of a differential form with a vector field).

(pseudo-) Laplacian on $p$ -forms. This is known as the Hodge Laplacian and also known as the Laplace–de Rham operator. It is defined by

$\displaystyle\Delta= d\delta+\delta d$ .

An important property of the Hodge Laplacian is that it commutes with the $*$ operator, i.e.

$\Delta * = * \Delta$ .

$\displaystyle\widetilde\Delta \alpha = {e^{ - 2f}}\left[ {\Delta \alpha - (n - 2p)d({\iota_{{\rm grad}f}}\alpha ) - (n - 2p - 2){\iota_{{\rm grad}f}}d\alpha + 2(n - 2p){\rm grad}f \wedge {\iota_{{\rm grad}f}}\alpha - 2{\rm grad}f \wedge \delta \alpha } \right]$ .

See also: Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1987.

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Ngô Quốc Anh

May 16, 2010

Conformal Changes of Riemannian Metrics

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