Projective Planes Over Division Algebras
In the special case n=2, Gromov's inequality becomes . This inequality can be thought of as an analog of Pu's inequality for the real projective plane . 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 is not its systolically optimal metric. In other words, the manifold admits Riemannian metrics with higher systolic ratio than for its symmetric metric, see Bangert et al. (2009).
Read more about this topic: Gromov's Inequality For Complex Projective Space
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