Let
(
M
,
g
)
(M,g)
be a Riemannian
n
n
-manifold, we denote by
R
i
c
Ric
and
S
c
a
l
Scal
the Ricci and the scalar curvatures of
g
g
. For each real number
k
>
n
k>n
, the modified Einstein tensors denoted
E
i
n
k
\mathrm {Ein}_k
is defined to be
E
i
n
k
≔
S
c
a
l
g
−
k
R
i
c
\mathrm {Ein}_k ≔Scal\, g -kRic
. Note that the usual Einstein tensor coincides with one half of
E
i
n
2
\mathrm {Ein}_2
and
E
i
n
0
=
S
c
a
l
.
g
\mathrm {Ein}_0=Scal.g
. It turns out that all these new modified tensors, for
0
>
k
>
n
0>k>n
, are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. The positivity of
E
i
n
k
\mathrm {Ein}_k
for some positive
k
k
implies the positivity of all
E
i
n
l
\mathrm {Ein}_l
with
0
≤
l
≤
k
0\leq l\leq k
and so we define a smooth invariant
E
i
n
(
M
)
\mathbf {Ein}(M)
of
M
M
to be the supremum of positive k’s that renders
E
i
n
k
\mathrm {Ein}_k
positive. By definition
E
i
n
(
M
)
∈
[
0
,
n
]
\mathbf {Ein}(M)\in [0,n]
, it is zero if and only if
M
M
has no positive scalar curvature metrics and it is maximal equal to
n
n
if
M
M
possesses an Einstein metric with positive scalar curvature. In some sense,
E
i
n
(
M
)
\mathbf {Ein}(M)
measures how far
M
M
is away from admitting an Einstein metric of positive scalar curvature.
In this paper, we prove that
E
i
n
(
M
)
≥
2
\mathbf {Ein}(M)\geq 2
, for any closed simply connected manifold
M
M
of positive scalar curvature and with dimension
≥
5
\geq 5
. Furthermore, for a compact
2
2
-connected manifold
M
M
with dimension
≥
6
\geq 6
and of positive scalar curvature, we show that
E
i
n
(
M
)
≥
3
\mathbf {Ein}(M)\geq 3
. We demonstrate as well that the invariant
E
i
n
(
M
)
\mathbf {Ein} (M)
of a manifold
M
M
increases after doing a surgery on
M
M
or by assuming that
M
M
has higher connectivity. We show that the condition
E
i
n
(
M
)
≤
n
−
2
\mathbf {Ein}(M)\leq n-2
does not imply any restriction on the first fundamental group of
M
M
. We define and prove similar properties for an analogous invariant namely
e
i
n
(
M
)
\mathbf {ein}(M)
. The paper contains several open questions.