Geometry of isoparametric submanifolds of Hilbert space and topology of spaces of curves with bounded curvature on surfaces.

• Grant number: 14/22556-3
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: February 01, 2015
• End date: January 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Claudio Gorodski
• Grantee: Pedro Paiva Zühlke d'Oliveira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 11/21362-2 - Group actions, submanifold theory and global analysis in Riemannian and pseudo-Riemannian geometry, AP.TEM

Abstract

We propose the investigation two non-related topics in the areas of differential geometry and geometric topology.The first topic involves the study of isoparametric submanifolds of a separable Hilbert space. The objective is to extend known results for the case of submanifolds of finite-dimensional Euclidean spaces to the case of infinite dimension. However, as there is no suitable classification of Hilbert-Lie groups nor of their representations, many of the methods that work in finite dimension cannot be employed. Our approach is to study the geometry of isoparametric submanifolds to attain results about their classification. We also propose to study the topology of spaces of curves with constrained curvature on complete manifolds (of finite dimension), especially on surfaces. These spaces possess extremely rich and surprising topological properties, and have been previously studied by other researchers, including the candidate.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ZUHLKE, PEDRO. On a class of immersions of spheres into space forms of nonpositive curvature. Geometriae Dedicata, v. 205, n. 1, p. 95-112, APR 2020. (14/22556-3)

SALDANHA, NICOLAU C.; ZUHLKE, PEDRO. Spaces of Curves with Constrained Curvature on Hyperbolic Surfaces. Indiana University Mathematics Journal, v. 69, n. 4, p. 1403-1443, 2020. (14/22556-3)