Ngô Quốc Anh

December 13, 2013

Norm of some 2-tensors involving the Ricci curvature tensor

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 4:09

Following the previous note where we discuss about the norm of the tensor \text{Ric}-\frac 1n \overline{\text{Scal}}g in terms of the trace-free Ricci tensor, i.e. the following identity

\displaystyle \boxed{{\left| {{\text{Ric}} - \dfrac{\overline{\text{Scal}}}{n}g} \right|^2} = |\mathop {{\text{Ric}}}\limits^ \circ {|^2} + \dfrac{1}{n}{(\text{Scal} - \overline{\text{Scal}})^2}}

holds. Today, we derive some further identites involving the Ricci tensor \text{Ric}. As we have already seen from the previous notes, the trace-free Ricci tensor is given by

\displaystyle {\mathop \text{Ric}\limits^ \circ} = \text{Ric} -\frac{g}{n}\text{Scal}.

First, we calculate \text{Ric}-\frac gn \text{Scal}. For simplicity, we denote by R and $S$ the Ricci tensor and the scalar curvature respectively. By definition, we have

\begin{array}{lcl} \displaystyle {\left| {R - \frac{g}{n}S} \right|^2} &=& \displaystyle {g^{im}}{g^{jn}}{\left( {R - \frac{g}{n}S} \right)_{ij}}{\left( {R - \frac{g}{n}S} \right)_{mn}} \hfill \\ &=& \displaystyle {g^{im}}{g^{jn}}\left( {{R_{ij}}{R_{mn}} - \frac{S}{n}({g_{ij}}{R_{mn}} + {g_{mn}}{R_{ij}}) + \frac{{{S^2}}}{{{n^2}}}{g_{ij}}{g_{mn}}} \right) \hfill \\ &=& \displaystyle {\left| R \right|^2} - \frac{S}{n}{g^{im}}{g^{jn}}({g_{ij}}{R_{mn}} + {g_{mn}}{R_{ij}}) + \frac{{{S^2}}}{{{n^2}}}{g^{im}}\underbrace {{g^{jn}}{g_{ij}}}_{\delta _i^n}{g_{mn}} \hfill \\ &=& \displaystyle {\left| R \right|^2} - \frac{S}{n}{g^{im}}\underbrace {{g^{jn}}{g_{ij}}}_{\delta _i^n}{R_{mn}} - \frac{S}{n}{g^{im}}\underbrace {{g^{jn}}{g_{mn}}}_{\delta _m^j}{R_{ij}} + \frac{{{S^2}}}{{{n^2}}}\underbrace {{g^{im}}{g_{mi}}}_n \hfill \\ &=& \displaystyle {\left| R \right|^2} - \frac{S}{n}\underbrace {{g^{nm}}{R_{mn}}}_S - \frac{S}{n}\underbrace {{g^{ij}}{R_{ij}}}_S + \frac{{{S^2}}}{n} \hfill \\ &=& \displaystyle {\left| R \right|^2} - \frac{{{S^2}}}{n}. \end{array}


December 12, 2013

An upper bound for the total integral of the Q-curvature in the non-negative Yamabe invariants

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 6:24

As we have already discussed once that a natural conformally invariant in dimension four is the following

\displaystyle Q_g=-\frac{1}{12}(\Delta\text{Scal}_g -\text{Scal}_g^2 +3|{\rm Ric}_g|^2)

which is commonly refered to the Q-curvature of metric g, see this topic. Note that, under a conformal change of the metric \widetilde g =e^{2u}g, the quantity Q transforms according to

\displaystyle 2Q_{\widetilde g}=e^{-4u}(P_gu+2Q_g)

where P=P_g denotes the Paneitz operator with respect to g. Keep in mind that the Paneitz operator is conformally invariant in the sense that

\displaystyle P_{\widetilde g}=e^{-4u}P_g

for any conformal metric \widetilde g =e^{2u}g. For any g, the operator P_g acts on a smooth function u on M via the following rule

\displaystyle {P_g}(u) = \Delta _g^2u + {\rm div}\left( {\frac{2}{3}\text{Scal}_g - 2{\rm Ric}_g} \right)du

which plays a similar role as the Laplace operator in dimension two. Observe that dv_{\widetilde g} = e^{4u}dv_g, therefore, a simple calculation shows

\displaystyle \int_M Q_{\widetilde g}dv_{\widetilde g}=\int_M Q_{\widetilde g}e^{4u}dv_g=\int_M Q_g dv_g.

Hence the total integral \int_M Q_g dv_g is conformally invariant.


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