A systolic inequality with remainder in the real projective plane

The first paper in systolic geometry was published by Loewner's student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric on the real projective plane. We prove a stronger version of Pu's systolic inequality with a remainder term.


Introduction
Loewner's systolic inequality for the torus and Pu's inequality [1] for the real projective plane were historically the first results in systolic geometry. Great stimulus was provided in 1983 by Gromov's paper [2] and later by his book [3].
Our goal is to prove a strengthened version with a remainder term of Pu's systolic inequality g sys 2 ( )≤ g area π 2 ( ) (for an arbitrary metric g on 2 ), analogous to Bonnesen's inequality L A R r 4π π 2 2 2 − ≥ ( − ), where L is the length of a Jordan curve in the plane, A is the area of the region bounded by the curve, R is the circumradius and r is the inradius.

The results
We define a closed three-dimensional manifold M 3 3 can be identified by differentiating the three defining equations of M along a path through v w , ( ). Thus, We define a Riemannian metric g M on M as follows. Given a point v w M , ( ) ∈ , let n v w = × and declare the basis n n w v 0, , , 0 , , to be orthonormal. This metric is a modification of the metric restricted to M from 3 3 6 × = . Namely, with respect to the Euclidean metric on 6 the above three vectors are orthogonal and the first two have length 1. However, the third vector has Euclidean length 2 , whereas we have defined its length to be 1.
⊆ ( ) denotes the span of n 0, ( ) and n,0 ( ), and then the metric g M on M is obtained from the Euclidean metric g on 6 (viewed as a quadratic form) as follows: .
Each of the natural projections p q M S , : ), is mapped by dp to the unit vector n TS v 2 ∈ . The same comments apply when the roles of p and q are reversed.
In the following proposition, integration takes place, respectively, over great circles C S 2 ⊆ , over the fibers in M, over S 2 , and over M. The integration is always with respect to the volume element of the given Riemannian metric. Since p and q are Riemannian submersions by Lemma 2.1, we can use Fubini's theorem to integrate over M by integrating first over the fibers of either p or q, and then over S 2 ; cf. Then, where equality in the second inequality occurs if and only if f is constant.
Proof. Using the fact that M is the total space of a pair of Riemannian submersions, we obtain  Then, , providing a probabilistic meaning for the quantity V f , as before.
By the uniformization theorem, every metric g on 2 is of the form g f g ) . Let f : 2 → + be such that g f g