Christophe Bavard | Searcharxiv

arXiv2016

Extensions maximales et classification des tores lorentziens munis d'un champ de Killing

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call "universal extensions". They are characterized by a condition of symmetry, the "reflexivity", and a by a rather weak completeness assumption, the absence of "saddles at infinity". Considering these surfaces as model spaces, we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, which is uniquely determined) in the following cases: compact surfaces and analytical surfaces. It allows us to give a classification of Lorentzian tori and Klein bottles with a Killing vector field.

math.DG↗

arXiv2014

Isodualité des réseaux euclidiens en petite dimension

We propose an algebraic and a geometric classification of euclidean isodual lattices of fixed rank. First, we prove that these lattices are distribued according to a finite number of algebraic types. Second, we show that they are parametrized by a finite number of symmetric spaces associated to the classical groups ${\bf SO}_0(p,q)$, ${\bf Sp}(2g,{\bf R})$ and ${\bf SU}(p,q)$. We obtain a complete discription of algebraic types and Gram matrices of isodual lattices up to rank 7. The maximal density problem is also discussed.

math.NT↗