A Riemannian manifold
(
M
n
,
g
)
(M^n,g)
is associated with a Schouten
(
0
,
2
)
(0,2)
-tensor
C
g
C_g
which is a naturally defined Codazzi tensor in case
(
M
n
,
g
)
(M^n,g)
is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional
F
k
[
g
]
=
∫
M
σ
k
(
C
g
)
d
v
o
l
g
\mathcal {F}_k[g]=\int _M\sigma _k(C_g)dvol_g
defined on
M
1
=
{
g
∈
M
|
V
o
l
(
g
)
=
1
}
\mathcal {M}_1=\{g\in \mathcal {M}|Vol(g)=1\}
, where
M
\mathcal {M}
is the space of smooth Riemannian metrics on a compact smooth manifold
M
M
and
{
σ
k
(
C
g
)
,
1
≤
k
≤
n
}
\{\sigma _k(C_g),\ 1\leq k\leq n\}
is the elementary symmetric functions of the eigenvalues of
C
g
C_g
with respect to
g
g
. We prove that if
n
≥
5
n\geq 5
and a conformally flat metric
g
g
is a critical point of
F
2
|
M
1
\mathcal {F}_2|_{\mathcal {M}_1}
with
F
2
[
g
]
≥
0
\mathcal {F}_2[g]\geq 0
, then
g
g
must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky’s very recent theorem that the critical point of
F
2
|
M
1
\mathcal {F}_2|_{\mathcal {M}_1}
with
F
2
[
g
]
≥
0
\mathcal {F}_2[g]\geq 0
characterized the three-dimensional space forms.