Loewner's torus inequality

Historically, the first lower bound for the volume of a Riemannian manifold in terms of a systole is due to Charles Loewner. In 1949, Loewner proved the first systolic inequality, in a course on Riemannian geometry at Syracuse University, cf. [Pu52]. Namely, he showed the following result.

Theorem 3.1 (C. Loewner)   Every Riemannian metric  ${\mathcal G}$ on the torus  ${\mathbb{T}}^2$ satisfies the inequality

$$\pisys _1({\mathcal G})^2\le \gamma_2\; \area ({\mathcal G}),$$ (3.1)

where  $\gamma_2=\frac{2}{\sqrt{3}}$ is the Hermite constant. A metric attaining the optimal bound (3.1) is necessarily flat, and is homothetic to the quotient of  ${\mathbb{C}}$ by the lattice spanned by the cube roots of unity.

For a proof, see [Ka07, Section 6.2].