On totally real submanifolds of some flat spaces

This work on totally real submanifolds of some flatspaces discusses specially the totally real submanifoldsin the flat Hermitian plane H2 where H2 is identified by4(E ; < , > ;J)~ The thesis contains three chapters:Chapter I deals with the introduction which containsthe definitions of differentiable manifolds, Riemannianmanifolds, complex manifolds, Hermitian manifolds, •.. etc.and the methods used in the following two chapters.Chapter 11 deals with totally real hypersurfaces inthe flat Hermitian plane H2 Theorem (1) states theinvariants of a real hypersurface Mc H2 .Theorem (2) shows the conditions necessary for a realhypersurface M C H2 to have a system of lines of curvatureand introduces the relations between the principal curvaturesand the invariants of the hypersurface M.The existence of a conformal mapping on a real hyper-surface MCH2 yields some relations stated in (2.2). In (2.3)the vanishing of a tangent vector field v on a real hyper-surface M C H2 is discussed under some assumptions.Theorem (3) states the vanishing of tangent vector fieldson a hypersphere M C H2.Theorem (4) shows the conditions necessary for thevanishing of tangent vector fields on a hypersurface MC H2