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

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

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

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.

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