Study Of Some Problems Of The Correlation Geometry

Classical geometry is extended to correlation geometry now, it contains the basic projective geometry and affine geometry. Much literature uses different axiomatic defi-nition, and it is difficult to prove the equivalence of these different axiomatic definition, which brings trouble to the comprehension and application of the results. Currently, the development trend of correlation geometry is geometry over ring, and "geometry transformation problem under the weaker condition," That’s simplify conditions of fundamental theorems of geometry and make it more perfect and easier application. Since the1990s, several scholars abroad characterized fundamental theorems of pro-jective geometry under the weaker conditions, that is,characterize mapping of the line mapped to the line between projective geometry over two division ring. On this basis, this paper will improve the theory of projective geometry and affine geometry in depth.This paper has three parts. In Chapter1, we introduce the topic background and research contents and main results.In Chapter2, we give an axiomatic definition of projective geometry, and discuss the equivalence definition, basic properties, the relation between projective geometry and vector space, morphisms of projective geometry. And we study the semi-linear map between the left (right) vector space over division ring. Later, some geometers abroad proved the fundamental theorems of projective geometry under the weaker conditions, we will improve the proof of the fundamental theorem to make it easier to read and understand. In the last section of this chapter, we discuss the theory of projective geometry over Bezout domain and the main results are gained by using the localization method over ring, we proved the following fundamental theorems of projective geometry over Bezout domain:letR’, R be two Bezout domains, and R be the division of let fractions of R, let V’=R’R’n,V=RRn, V=RR, n≥3. Assume that g:p(V’)â†'P(V) is a non-degenerate morphism. Then there exists a canonical morphism Ï„:Vâ†'V such that Ï„ o g is a non-degenerate morphism from P(V) to P(V),and Ï„ o g((x))=<xσP>,(?)x∈V’, where P∈Rn×n is fixed, and σâ†'R is a ring homomorphism with1R’σ=1R,OR’σ=0R.Chapter3introduce the theory of affine geometry, we discuss the morphism-s of affine geometry and the theory of weighted half-affine map. The following is fundamental theorem of affine geometry under the weaker conditions:LetV and V’be left (or right)vector spaces over division rings D and D’, and their dimensions≥2. where|D|≥4. then every non-degenerate morphism f:Vâ†'V’ is a weighted semi-affine map. This is an important result, but it’s not detailed in literature, This paper improve the proof of this theorem and also give specific formula of weighted semi-affine map for easier understand and application.