Submanifolds and Submersions in Finsler Geometry

• Grant number: 24/03446-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: May 01, 2024
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marcos Martins Alexandrino da Silva
• Grantee: Patrícia Marçal
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 22/16097-2 - Modern methods in differential geometry and geometric analysis, AP.TEM

Abstract

The theory of Finsler submanifolds has progressed at slower pace then its counterpart, Riemannian submanifolds. This might be partly because there is no natural way to induce geometric objects to subspaces, or partly due to calculations in coordinates quickly becoming cumbersome. Often, progress is pushed by the prospect of applications, specially in Physics and Biology. Since the introduction of isometric submersions for Finsler spaces in 2001, new techniques can be developed and employed to investigate Finsler submanifolds. With this project, we wish to address the following topics: elaboration of a survey on Finsler submersions and foliations, the study of isoparametric hypersurfaces, submersions, and foliations of some prominent classes of Finsler metrics, such as those arising from navigation, warped products, $(\alpha, \beta)$-metrics, and homogeneous spaces. We will also seek to develop some subjects that are directly related to the topics above; namely: submersions or foliations with some constant mean curvature, mean curvature flow, and Morse-Bott functions.