The Cheeger-Gromoll splitting theorem and applications

Author(s)

Danielle Velloso Ferreira

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2025-02-07

Examining board members

Claudio Gorodski; Ruy Tojeiro de Figueiredo Junior; Dirk Toben

• Advisor: Claudio Gorodski

Abstract

In Riemannian geometry there are several results about the structure of manifolds depending on their curvature. The Cheeger-Gromoll Splitting Theorem is an example of this type of result. The theorem allows us to decompose a manifold as a Riemannian product, one of the factors being a Euclidean space. The objective of this dissertation is to present a detailed proof of the Cheeger-Gromoll Splitting Theorem following the arguments presented in the article by Eschenburg and Heintze. We chose to follow this alternative demonstration, as it uses only basic tools from Riemannian geometry and Calculus, while the original demonstration proposed by Cheeger and Gromoll uses the theory of regularity of elliptic functions, which is a more advanced result from PDE. Furthermore, at the end of this dissertation we will present the proofs of two results that use the Splitting Theorem: the Alekseevskii and Kimel\'fel\'d Theorem and an algebraic characterization of the fundamental group of compact manifolds. (AU)

• FAPESP's process: 22/13818-0 - The Cheeger-Gromoll Splitting Theorem and Applications
• Grantee: Danielle Velloso Ferreira
• Support Opportunities: Scholarships in Brazil - Master