Introduction to differential forms and applications

• Grant number: 14/19034-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2014
• End date: November 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Glaucio Terra
• Grantee: Rodrigo Rey Carvalho
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

Our aim is to study differential forms in Differential Geometry and applications. We shall consider vector-valued differential forms (taking values on Banach spaces) defined on open subsets of Banach spaces or finite dimensional differentiable manifolds. The applications to be considered are: 1) elementary theory of analytic functions of one complex variable by means of a differential forms approach; 2) Cartan method of the moving frame and applications in Differential Geometry, such as Gauss-Bonnet and Morse theorems in the context of embedded surfaces in R3; 3) geometric version of local Frobenius theorem for total differential equations, using differential forms. The student is expected to contribute with his own ideas and proofs.