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Systole and Hermite invariant. (Systole et invariant d’Hermite.) (French) Zbl 1011.53035

For a lattice \(\Lambda\) of determinant 1 in the Euclidean space \({\mathbb E}^n\) the Hermite invariant is defined by \(\mu(\Lambda)=\text{ inf} \left\{\langle x,x\rangle; x\in\Lambda,x\neq 0\right\}\). The systole of a compact Riemannian manifold \(M\), defined as the minimal length of a non-contractible curve in \(M\), is a far reaching generalization of this concept, because \(\mu(\Lambda)\) appears as the systole of the flat torus \({\mathbb E}^n/\Lambda\).
The author introduces a new unifying geometric framework for certain generalisations of the notion of the Hermite invariant, namely, flat tori, principally polarized abelian varieties of dimension \(g\), Riemannian surfaces of genus \(g\) endowed with the Poincaré metric. All these objects can be considered as points of certain Riemannian manifolds, namely of the space of \(P_n\) of Gram matrices (i.e., positive symmetric matrices of determinant 1), of the Siegel domain \({\mathcal H}_g\), and of the Teichmüller space \(T_g\), endowed with the Weil-Petersson metric, respectively.
The generalizations of the Hermite invariant appear as the minimum of a family of length functions on the corresponding parameter space. This permits to develop for each of these cases the theory in analogy to the case of lattices, which serves as guide. In the first chapter a geometric study of Hermite’s invariant is given. The classical notions of the theory of lattices are transferred to geometrical notions on the space \(P_n\). A geometric interpretation of the length function of lattices as exponentials of the Busemann function is given.
In chapter II the key notions ’perfection’ and ’eutaxy’, due to Voronoï, are defined with the help of gradients of length functions and used to characterise local maxima of the Hermite invariant. For the case of Riemannian surfaces the following analogue of Voronoï’s theorem is obtained: A Riemannian surface is a local maximum of the systole if and only if it is perfect and eutactic. In the last part it is observed that for principally polarized abelian varieties the situation changes: Eutactic principally polarized abelian varieties are not isolated and the Hermite invariant is not a Morse function on \({\mathcal H}_g\).

MSC:

53C20 Global Riemannian geometry, including pinching
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
14K99 Abelian varieties and schemes
53C22 Geodesics in global differential geometry
53C35 Differential geometry of symmetric spaces
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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