Flavor
To give a preliminary idea of the flavor of the field, one could make the following observations. The main thrust of Thom's remark to Berger quoted above appears to be the following. Whenever one encounters an inequality relating geometric invariants, such a phenomenon in itself is interesting; all the more so when the inequality is sharp (i.e., optimal). The classical isoperimetric inequality is a good example.
In systolic questions about surfaces, integral-geometric identities play a particularly important role. Roughly speaking, there is an integral identity relating area on the one hand, and an average of energies of a suitable family of loops on the other. By the Cauchy-Schwarz inequality, energy is an upper bound for length squared; hence one obtains an inequality between area and the square of the systole. Such an approach works both for the Loewner inequality
for the torus, where the case of equality is attained by the flat torus whose deck transformations form the lattice of Eisenstein integers,
and for Pu's inequality for the real projective plane P2(R):
- ,
with equality characterizing a metric of constant Gaussian curvature.
An application of the computational formula for the variance in fact yields the following version of Loewner's torus inequality with isosystolic defect:
where f is the conformal factor of the metric with respect to a unit area flat metric in its conformal class. This inequality can be thought of as analogous to Bonnesen's inequality with isoperimetric defect, a strengthening of the isoperimetric inequality.
A number of new inequalities of this type have recently been discovered, including universal volume lower bounds. More details appear at systoles of surfaces.
Read more about this topic: Systolic Geometry
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