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This study is about immersions of surfaces in a four dimensional space with a non deqenerated bilinear form. In the cases studied, this form induces in the normal and tangent planes, forms of type (+,+) or (+,-). Three situations: Y₁ , Y₂ and Y₃ are studied and they are based on two papers: one of John A. Little [4] and the other of P. F. J. Dhooghe [3]. In chapter one we see the requirements we need to fix the notations and study the theory involving the second fundamental form too. In chapter two we speak about the cross fields and thenotion of singular point of a cross field. We consider topics in geometrical transversality like models singularities and generic immesions. In chapter three, we consider some cross fields that give informations about the singularities and study the case Y₃ separetely because there aren't cross fields there. Át the end, we come to global theorems. Our main purpose was a comparative study that ballowed us to elaborate some theorems e to prove others inexistents in the basic papers. We elucidated some ambiguous topics in [3] and because the situation there wasn't so usual, we had to be careful and decided to give the proofs of many results considering the cases Y₂ and Y₃. A result to be pointed out is that when we consider the bilinear form Y² in E⁴ , the corollary 1.22, given in [4], becomes true, in spite of it isn't correct when we consider the bilinear form Y₁ in E⁴ (see example given by Asperti in [I]).