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Generalized degree and optimal Loewner-type inequalities. (English) Zbl 1067.53031
The results presented in the reviewed paper are generalizations of optimal inequalities of C. Loewner and M. Gromov. The unpublished Loewner inequality (circa 1949) relates the length of the shortest noncontractible loop of an arbitrary Riemannian metric on the 2-torus to its surface area. Half a century later M. Gromov obtained an optimal generalization of Loewner’s inequality for a compact manifold of equal dimension and first Betti number.
The authors of this paper prove lower bounds for the total volume in terms of the homotopy systole and the stable systole. Their main tool is the construction of an area-decreasing map to the Jacobi torus. It turns out that one can succesfully combine this construction with the coarea formula, yielding new optimal inequalities. This paper includes also interesting historical remarks and a discussion of the related systolic literature.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
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