On a class of immersions of spheres into space forms of nonpositive curvature

Author(s)

Zuhlke, Pedro

• Document type: Journal article
• Source: Geometriae Dedicata; v. 205, n. 1, p. 95-112, APR 2020.
• Web of Science Citations: 0

Abstract

Let Mn+1 (n = 2) be a simply-connected space form of sectional curvature -.2 for some. = 0, and I an interval not containing {[}-.,.] in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere Sn. These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space F of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of Sn. An equivalence assigns to each element of F a suitable modification of its Gauss map. For M not simply-connected, F is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M. (AU)

• FAPESP's process: 14/22556-3 - Geometry of isoparametric submanifolds of Hilbert space and topology of spaces of curves with bounded curvature on surfaces.
• Grantee: Pedro Paiva Zühlke D'Oliveira
• Support Opportunities: Scholarships in Brazil - Post-Doctoral