Symplectic and orthogonal geometry
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This thesis treats metric structures and transformations of finite dimensional spaces over commutative fields. The definition of metric structures of spaces is given in the first section of the chapter I. How such metric structures can be described in terms of bases and how the expressions depend upon the choices of bases are then studied. The geometries where XY = 0 implies YX = 0 for any elements X,Y of the space are sought in section 2. The orthogonal and symplectic geometries are introduced as the only two types of geometries which satisfy this condition. In connection with metric structures of spaces, bilinear and quadratic forms are discussed in section 3. Metric structures of orthogonal geometries are shown to be determined by symmetric bilinear forms and those of symplectic geometries are determined by skew symmetric bilinear forms. A study of orthogonal geometries is shown·to be equivalent to a study of quadratic forms. Irreducible subspaces of orthogonal and symplectic geometries are discussed in section 4, by considering the properties of kernels, radicals and singularities of spaces and defining isotropic spaces and vectors. As a result one can see that an orthogonal geometry is an orthogonal sum of lines and that a nonsingular symplectic space is an orthogonal sum of hyperbolic planes. For example, a Euclidean space is an orthogonal sum of lines generated by the characteristic vectors of the matrix describing its metric structure. Reducing bilinear forms to its canonical forms, one can see that there exists only one non-singular symplectic space, if a dimension and a field are given. The subject of chapter II is mappings of spaces on which metric structures are defined. The definition of homomorphisms and isometries are first given in section_l and 2, together with their matrix representations relative to bases of spaces. Rotations and reflexions are then introduced as two types of isometries. In section 3, the definition of involutions is given. It then follows that every involution has a form -1u perpendicular lw, where U and W are mutually orthogonal subspaces. Some important properties of isometries on orthogonal geometries are stated in section 4. As examples, isometries on two- and three-dimensional Euclidean spaces and Lorentz transformations on two dimensional space are discussed in section 5 and 6. Finally orthogonal geometries of n-dimensional spaces over finite fields are discussed, together with an example of geometric structures and isometries of a two-dimensional orthogonal space over a field J3.
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