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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[compact|0TVJZYB6SH]] connected [[Riemannian manifold|7CBT4SM83Y]] that is not simply connected.\nA closed curve in $M$ is *noncontractible* if it is not freely homotopic to a constant curve.\nThe *systole* $\\operatorname{sys}(M, g)$ is the infimum of the lengths of the noncontractible closed curves in $M$; it is attained and positive.\nIf $M$ has dimension $n$, the *systolic ratio* is $\\operatorname{sys}(M, g)^n / \\operatorname{vol}(M, g)$.\n*Systolic geometry* is the study of the systole and of upper bounds for the systolic ratio over all metrics $g$ on a fixed manifold $M$.\n\n## Remarks\n\nExample: the flat torus $\\R^2 / \\Z^2$ has systole $1$, attained by the images of the coordinate segments, and area $1$, so its systolic ratio is $1$.\n[[Loewner's torus inequality|1Y9X1YW6GH]] says that the systolic ratio of any metric on the torus is at most $2/\\sqrt{3}$.\n\nNon-example: on the round sphere $S^2$ every closed curve is contractible, so the systole is not defined.\n\nThe systolic ratio does not change when $g$ is multiplied by a positive constant.\nThe same words are used for higher systoles, defined with cycles of higher dimension instead of curves.\n",
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