In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
s
t
s
y
s
2
n
≤
n
!
v
o
l
2
n
(
C
P
n
)
,
valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here
s
t
s
y
s
2
is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line
C
P
1
⊂
C
P
n
in 2-dimensional homology.
The inequality first appeared in Gromov (1981) as Theorem 4.36.
The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.
In the special case n=2, Gromov's inequality becomes
s
t
s
y
s
2
2
≤
2
v
o
l
4
(
C
P
2
)
. This inequality can be thought of as an analog of Pu's inequality for the real projective plane
R
P
2
. In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on
H
P
2
is not its systolically optimal metric. In other words, the manifold
H
P
2
admits Riemannian metrics with higher systolic ratio
s
t
s
y
s
4
2
/
v
o
l
8
than for its symmetric metric (Bangert et al. 2009).
