PERFECT FLUIDS AND SIGMA MODELS: A MATHEMATICAL INVESTIGATION BETWEEN GENERAL RELATIVITY AND COTTON GRAVITY

In this thesis, we deal with two physics-inspired structures on Riemannian manifolds, depending on a smooth map φ between Riemannian manifolds and several other functions and constants; they are called Cotton φ-Perfect Fluids (C-φ-PF) and φ-static perfect fluids space-times (φ-SPFST). The latter are special solutions of the Einstein field equations on a static space-time with matter sources modelled by a perfect fluid and a map φ, while the former are the natural generalization of φ-SPFSTs in the framework of a gravitational theory of recent introduction, called Cotton Gravity. We highlight the role of the Introduction in clarifying the physical background of this work and the expected contributions that such structures might give to Riemannian Geometry. In Chapter 1 we fix the notations and conventions and review some of the basics of Riemannian Geometry and General Relativity. We begin Chapter 2 by introducing the theory of Cotton Gravity and showing how it preserves the solutions of General Relativity. We then derive the equations of C-φ-PF and show their variational origin; the class of variations that we need to consider is such that the connection, instead of the metric, is the fundamental entity. To obtain the main result of this chapter, a rigidity result for a C-φ-PF under some curvature conditions which are inspired by the theory of Ricci solitons, one needs to face a novel and specific property of Cotton Gravity, that of depending on third order derivatives of the metric tensor. In doing so, we will unveil the special relevance that Codazzi tensors hold in this theory, as well as the important relations between the equations of C-φ-PF and the first integrability condition of a φ-SPFST. Inspired by some works in conformal geometry, we then conclude this chapter by studying the obstruction to a C-φ-PF to be a φ-SPFST and relating it to some special algebraic properties of the Weyl tensor. In Chapter 3 we deal with φ-SPFSTs. We derive other rigidity results under new curvature condi tions, for both manifolds with and without boundary. The proof in the two cases are surprisingly different, but both rely on some special integral identities that make our assumptions effective. We then prove an Obata-type theorem that characterizes the standard spheres as the only compact manifolds supporting a closed, conformal vector field and a structure of generalized Harmonic Einstein manifold; we apply it to deduce a rigidity result for a closed φ-SPFST for which a suitable modification of the Schouten tensor is Codazzi and has some elementary symmetric function in its eigenvalues which is positive and constant.