On the topology of closed manifolds with quasi-constant sectional curvature

Funar, Louis

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 367-423.

Nous montrons que les variétés fermées admettant une métrique générique dont la courbure sectionnelle est localement quasi-constante sont des sommes graphées de variétés de courbure constante. Ensuite nous étendons ce résultat au cas des espaces QC dont l’ensemble des points isotropes pourrait être arbitraire en démontrant que, sous une condition de positivité et lorsque leurs groupes fondamentaux sont sans torsion, ils sont difféomorphes à des sommes connexes de fibrés en sphères sur le cercle.

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups, they are still diffeomorphic to connected sums of spherical bundles over the circle.

Classification : 53C21, 53C23, 53C25, 57R42
Keywords: Curvature, conformal geometry, topology, curvature leaves, codimension-one isometric immersions, foliations, second fundamental form
Mot clés : Courbure sectionnelle, géométrie conforme, topologie, feuilles de courbure, immersions isométriques en codimension 1, feuilletages, deuxième forme fondamentale

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