Ειδικές επιφάνειες του χώρου Ε3 1 με ΔIII r = A r και διαρμονικές υπερεπιφάνειες Μ3 2 του χώρου Ε4 2

Abstract

In the present Ph.D. Thesis we study three problems referred in the pseudo-Euclidean geometry. In the ¯rst two chapters, Chapter 1 and Chap- ter 2 we review known results and describe the basic notions of the Rie- mannian and pseudo-Riemannian geometry. In Chapter 3 we study surfaces of revolution of the 3-dimensional Lorentz-Minkowski space satisfying given geometric condition. In Chapter 4 we ¯nd all the canonical forms of the shape operator of the 3-dimensional hypersurfaces of signature (-, +, -) of the 4-dimensional pseudo-Euclidean space of signature (-, +, -, +). Finally in Chapter 5 we study the relation which exists between the biharmonic and minimal hypersurfaces referred in Chapter 4, by using their shape operator. More precisely we prove that every such biharmonic hypersurface is minimal. The main results of this Ph.D thesis are published (or has been accepted for publication) in the papers of Petoumenos et al (Surfaces of Revolution in the 3-dimensional Lorentz-Minkowski space E3 1 satisfying ¢III~r = A~r, Bull. Greek Math. Soc., 2005), Petoumenos et al (On the shape operator of the hypersurfaces M3 2 of E4 2 , Bull. Greek Math. Soc., 2007) and Petoumenos et al (Biharmonic hypersurfaces of type M3 2 in E4 2 , Houston J. Math., 2009).