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Microorganisms are single-celled organisms are small, not observed with the naked eye but to use a microscope. The term microbial does not correspond to any taxon in scientific classification. It includes viruses, bacteria, archaea, fungi, microscopic algae, protozoa. Etc. | INDUCED GEOMETRY ON R4n NGUYEN VIET HAI a Abstract. The present paper is a continuation of Nguyen Viet Hai s ones 4 5 . In this the author give a method to construct hypersymplectic structures on R4n from affine-symplectic data on R2n. 1 PRELIMINARIES A hypersymplectic structure on a 4n-dimensional manifold M is given by J E Ố where J E are endomorphisms of the tangent bundle of M such that J2 -1 E2 1 JE -EJ Ố is a neutral metric that is of signature 2n 2n satisfying Ỗ X Y Ỗ JX JY -Ỗ EX EY for all vector fields X Y on M and the following associated 2-forms are closed W1 X Y Ỗ JX Y U2 X Y ỗ EX Y u i X Y Ỗ JEX Y . In 5 we have determined the flat torsion-free connections on the 2-dimensional Lie algebras which are compatible with a symplectic form and obtained their equivalence classes. We showed all the flat torsion-free connections that preserve a sym-plectic form on the 2-dimensional Lie algebras namely on R2 and on aff R . Those importante results used in the 4-dimensional case. In 4 we presented a method to contruct four-dimensional Lie algebras carrying a hypersymplectic structure from two 2-dimensional Lie algebras equipped with compatible flat torsion-free connections and symplectic forms. Using this method we obtained the classification up to equivalence of all left-invariant hypersymplectic structures on 4-dimensional Lie groups. All those Lie groups are exponential type. The purpose of this paper is to give a procedure to construct hypersymplectic structures on R4n with complete and not necessarily flat associated neutral metrics. 1 Nhận bài ngày 24 11 2006. sửa chữa xong ngày 14 12 2006. The idea behind the construction will be to consider the canonical flat hypersymplec-tic structure on R4n and then translate it by using an appropriate group acting simply and transitively on R4n. This group will be a double Lie group R4n R2n x 0 0 x R2n constructed from affine data on R2n. The paper is organized as follows. In 2 we give to R4n a structure of a nilpotent
