Ngô Quốc Anh

May 16, 2010

Conformal Changes of Riemannian Metrics

Filed under: Riemannian geometry — Tags: — 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

  1. (pointwise) conformal if there exists a C^\infty function f on M such that

    \displaystyle \widetilde g=e^{2f}g;

  2. 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}.

So

\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}.

So

\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.

So

\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.

So

\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 .

So

\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.

So

\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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