A systolic inequality with remainder in the real projective plane

1 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 sys 2 ( g ) ≤ π 2 area ( g ) (for an arbitrary metric g on ℝ ℙ 2 ), analogous to Bonnesen’s inequality L 2 − 4 π A ≥ π 2 ( R − r ) 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.

Note that both the original proof in Pu ([1], 1952) and the one given by Berger ([4], 1965, pp. 299–305) proceed by averaging the metric and showing that the averaging process decreases the area and increases the systole. Such an approach involves a five-dimensional integration (instead of a three-dimensional one given here) and makes it harder to obtain an explicit expression for a remainder term. Analogous results for the torus were obtained in ref. [5] with generalizations in ref. [6,7,8,9,10,11,12,13,14,15,16,17].

2 The results

We define a closed three-dimensional manifold M ⊆ ℝ 3 × ℝ 3 by setting

where v ⋅ w is the scalar product on ℝ 3 . We have a diffeomorphism M → S O ( 3 , ℝ ) , ( v , w ) ↦ ( v , w , v × w ) , where v × w is the vector product on ℝ 3 . Given a point ( v , w ) ∈ M , the tangent space T ( v , w ) M 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 ( 0 , n ) , ( n , 0 ) , ( w , − v ) of T ( v , w ) M to be orthonormal. This metric is a modification of the metric restricted to M from ℝ 3 × ℝ 3 = ℝ 6 . Namely, with respect to the Euclidean metric on ℝ 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. Thus, if A ⊆ T ( v , w ) M denotes the span of (0, n ) and ( n ,0) , and B ⊆ T ( v , w ) M is spanned by ( w , − v ) , then the metric g M on M is obtained from the Euclidean metric g on ℝ 6 (viewed as a quadratic form) as follows:

Each of the natural projections p , q : M → S 2 given by p ( v , w ) = v and q ( v , w ) = w exhibits M as a circle bundle over S 2 .

The projection p maps the fiber q − 1 ( w ) onto a great circle of S 2 . This map preserves length since the unit vector ( n ,0) , tangent to the fiber q − 1 ( w ) at ( v , w ) , is mapped by dp to the unit vector n ∈ T v S 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. [18, Lemma 4]. By the remarks above, if C = p ( q − 1 ( w ) ) and f : S 2 → ℝ , then ∫ q − 1 ( w ) f ∘ p = ∫ C f .

where equality in the second inequality occurs if and only if f is constant.

proving the second inequality. Here, equality occurs if and only if f and 1 are linearly dependent, i.e., if and only if f is constant.□

We define the quantity V f by setting V f = ∫ S 2 f 2 − 1 4 π ∫ S 2 f 2 . Then, Proposition 2.2 can be restated as follows.

We can assign a probabilistic meaning to V f as follows. Divide the area measure on S 2 by 4 π , thus turning it into a probability measure μ . A function f : S 2 → ℝ + is then thought of as a random variable with expectation E μ ( f ) = 1 4 π ∫ S 2 f . Its variance is thus given by

The variance of a random variable f is non-negative, and it vanishes if and only if f is constant. This reproves the corresponding properties of V f established above via the Cauchy-Schwarz inequality.

Now let g 0 be the metric of constant Gaussian curvature K = 1 on ℝ ℙ 2 . The double covering ρ : S 2 → ( ℝ ℙ 2 , g 0 ) is a local isometry. Each projective line C ⊆ ℝ ℙ 2 is the image under ρ of a great circle of S 2 .

For ℝ ℙ 2 we define V ¯ f = ∫ ℝ ℙ 2 f 2 − 1 2 π ∫ ℝ ℙ 2 f 2 = 1 2 V f ∘ ρ . We obtain the following restatement of Proposition 2.4.

Relative to the probability measure induced by 1 2 π g 0 on ℝ ℙ 2 , we have E ( f ) = 1 2 π ∫ ℝ ℙ 2 f , and therefore Var ( f ) = 1 2 π V ¯ f , providing a probabilistic meaning for the quantity V ¯ f , as before.

By the uniformization theorem, every metric g on ℝ ℙ 2 is of the form g = f 2 g 0 , where g 0 is of constant Gaussian curvature + 1 , and the function f : ℝ ℙ 2 → ℝ + is continuous. The area of g is ∫ ℝ ℙ 2 f 2 , and the g-length of a projective line C is ∫ C f . Let L be the shortest length of a noncontractible loop. Then, L ≤ m ¯ where m ¯ is defined in Proposition 2.4, since a projective line in ℝ ℙ 2 is a noncontractible loop. Then, Corollary 2.5 implies area ( ℝ ℙ 2 , g ) − 2 L 2 π ≥ V ¯ f ≥ 0 . If area ( ℝ ℙ 2 , g ) = 2 L 2 π , then V ¯ f = 0 , which implies that f is constant, by Corollary 2.5. Conversely, if f is a constant c, then the only geodesics are the projective lines, and therefore, L = c π . Hence, 2 L 2 π = 2 π c 2 = area ( ℝ ℙ 2 ) . We have thus completed the proof of the following result strengthening Pu’s inequality.

where the variance is with respect to the probability measure induced by 1 2 π g 0 . Furthermore, equality area ( g ) = 2 L 2 π holds if and only if f is constant.