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Research report: "Geometry induced on R4n"

Geometry is a branch of mathematics related research space. Using experience, or perhaps by intuition, it is recognized by the space fundamental characteristics, the geometric axioms called the system. Axiomatic system including the original concept is not defined and the axioms (also known as the proposition) does not prove a relationship defined between the concepts. | 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