We want to find the necessary condition of being the critical points of
on four dimensional manifold.
1. Preliminary
Suppose
is a Riemannian manifold with
.
is the Schouten tensor

and denote
. Define
![\displaystyle \sigma_2(g)=\frac{1}{2}[(\text{\,Tr\,} P_g)^2-|P_g|_g^2] \displaystyle \sigma_2(g)=\frac{1}{2}[(\text{\,Tr\,} P_g)^2-|P_g|_g^2]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csigma_2%28g%29%3D%5Cfrac%7B1%7D%7B2%7D%5B%28%5Ctext%7B%5C%2CTr%5C%2C%7D+P_g%29%5E2-%7CP_g%7C_g%5E2%5D&bg=ffffff&fg=000000&s=0)

where
. It is well known that
is conformally invariant.
Suppose
where
is a symmetric 2-tensor. We want to calculate the first derivative of
at
. To that end, let us list some basic facts (see the book of Toppings). Firstly denote
the divergence operator and


where
is the Lichnerowicz Laplacian. Then the first variation of Ricci curvature and scalar curvature are



where we were using upper dot to denote the derivative with respect to
.
2. First variation of the sigma2 functional
Lemma 1
is a critical point of
if and only if

where
.
Proof:

where
. Since we have
![\displaystyle \int_M\langle P,\dot P\rangle\\ =\frac{1}{n-2}\int_M\langle P, \dot Ric-\dot Jg-Jh\rangle =\frac{1}{n-2}[\langle P, \dot Ric\rangle-\dot J J-J\langle h,P\rangle] \displaystyle \int_M\langle P,\dot P\rangle\\ =\frac{1}{n-2}\int_M\langle P, \dot Ric-\dot Jg-Jh\rangle =\frac{1}{n-2}[\langle P, \dot Ric\rangle-\dot J J-J\langle h,P\rangle]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_M%5Clangle+P%2C%5Cdot+P%5Crangle%5C%5C+%3D%5Cfrac%7B1%7D%7Bn-2%7D%5Cint_M%5Clangle+P%2C+%5Cdot+Ric-%5Cdot+Jg-Jh%5Crangle+%3D%5Cfrac%7B1%7D%7Bn-2%7D%5B%5Clangle+P%2C+%5Cdot+Ric%5Crangle-%5Cdot+J+J-J%5Clangle+h%2CP%5Crangle%5D+&bg=ffffff&fg=000000&s=0)
![\displaystyle =\frac{1}{n-2}[-\frac{1}{2}\langle h,\Delta_L P\rangle+\langle h,Hess(J)\rangle-\frac{1}{2}\Delta J \text{\,Tr\,} h-\dot J J-J\langle h,P\rangle] \displaystyle =\frac{1}{n-2}[-\frac{1}{2}\langle h,\Delta_L P\rangle+\langle h,Hess(J)\rangle-\frac{1}{2}\Delta J \text{\,Tr\,} h-\dot J J-J\langle h,P\rangle]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D%5Cfrac%7B1%7D%7Bn-2%7D%5B-%5Cfrac%7B1%7D%7B2%7D%5Clangle+h%2C%5CDelta_L+P%5Crangle%2B%5Clangle+h%2CHess%28J%29%5Crangle-%5Cfrac%7B1%7D%7B2%7D%5CDelta+J+%5Ctext%7B%5C%2CTr%5C%2C%7D+h-%5Cdot+J+J-J%5Clangle+h%2CP%5Crangle%5D+&bg=ffffff&fg=000000&s=0)
Plugging this into the derivative of
to get


In order to simplify the above equation, we recall the definition of Lichnerowicz Laplacian 


Apply (1) to get
![(n-1)\dot J J=\frac12J\dot R=\frac12J[\delta^2h-\Delta(\text{\,Tr\,} h)-\langle h,Ric\rangle] (n-1)\dot J J=\frac12J\dot R=\frac12J[\delta^2h-\Delta(\text{\,Tr\,} h)-\langle h,Ric\rangle]](https://s0.wp.com/latex.php?latex=%28n-1%29%5Cdot+J+J%3D%5Cfrac12J%5Cdot+R%3D%5Cfrac12J%5B%5Cdelta%5E2h-%5CDelta%28%5Ctext%7B%5C%2CTr%5C%2C%7D+h%29-%5Clangle+h%2CRic%5Crangle%5D&bg=ffffff&fg=000000&s=0)
![=\frac{1}{2}[\langle h, Hess(J)\rangle-\text{\,Tr\,} h\Delta J-(n-2)J\langle h, P\rangle-J^2\text{\,Tr\,} h] =\frac{1}{2}[\langle h, Hess(J)\rangle-\text{\,Tr\,} h\Delta J-(n-2)J\langle h, P\rangle-J^2\text{\,Tr\,} h]](https://s0.wp.com/latex.php?latex=%3D%5Cfrac%7B1%7D%7B2%7D%5B%5Clangle+h%2C+Hess%28J%29%5Crangle-%5Ctext%7B%5C%2CTr%5C%2C%7D+h%5CDelta+J-%28n-2%29J%5Clangle+h%2C+P%5Crangle-J%5E2%5Ctext%7B%5C%2CTr%5C%2C%7D+h%5D+&bg=ffffff&fg=000000&s=0)
Therefore we can simplify it to be


Let us denote
. Using the fact
and the definition of
,

where


Remark 1 It is easy to verify
, this is equivalent to say
is invariant under conformal change. More precisely, letting
, then

Remark 2 If
is an Einstein metric with
, then
,
and

It is easy to verify that
. In other words, Einstein metrics are critical points of
.
Are there any non Einstein metric which are critical points of
?
Here is one example. Suppose
, where
is the sphere with standard round metric and
is a two dimensional compact manifolds with sectional curvature
.
is endowed with the product metric. We can prove
,
,
,
and consequently
.
Note that the above example is a locally conformally flat manifold. For this type of manifold, we have the following lemma which can say
Lemma 2 Suppose
is locally conformally flat and
, then
Proof: When
is locally conformally flat,

is equivalent to

Actually this is equivalent to the Bach tensor
is zero. 
3. Another point of view
We have the Euler Characteristic formula for four dimensional manifolds

therefore the critical points for
will be the same as the critical points of
. However, the functional

is well studied by Bach. The critical points of this functional satisfy Bach tensor equal to 0.

Obviously,
for Einstein metric, but not all Bach flat metrics are Einstein. For example
for any locally conformally flat manifolds.