Loewner's torus inequality

In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.

In 1949 Charles Loewner proved that every metric on the 2-torus ${\displaystyle \mathbb {T} ^{2}}$ satisfies the optimal inequality

where "sys" is its systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant ${\displaystyle \gamma _{2}}$ in dimension 2, so that Loewner's torus inequality can be rewritten as

The inequality was first mentioned in the literature in Pu (1952).

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