Subspace operations in affine Klingenberg spaces
نویسندگان
چکیده
In two previous papers we introduced the notion of an Affine Klingenberg space A and presented a geometric description of its free subspaces. Presently, we consider the operations of join, intersection and parallelism on the free subspaces of A. As in the case of ordinary affine spaces, we obtain the Parallel Postulate. The situation with join and intersection is not that straightforward. In particular, the central problem is whether the join of two free subspaces is free? We show that if A is not an ordinary affine space and dim A ≥ 4 then A has a subspace which is both not free and the join of two free subspaces. Thus, join and intersection do not possess the usual closure properties. We determine necessary and sufficient conditions under which the join of two free subspaces is free, and in such a case we verify the Dimension Formula. The subspace operations are essential tools for establishing when A is desarguesian and when it can be embedded in a projective Klingenberg space.
منابع مشابه
Convergence of Minimum Norm Elements of Projections and Intersections of Nested Affine Spaces in Hilbert Space
We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and in...
متن کاملA Note on the Structure of Affine Subspaces of
This paper investigates the structure of general affine subspaces of ( ) L . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a ...
متن کاملProof of the Carathéodory Conjecture by Mean Curvature Flow in the Space of Oriented Affine Lines Brendan Guilfoyle and Wilhelm Klingenberg
متن کامل
A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition
متن کامل
Affine Spaces within Projective Spaces
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finitedimensional or pappian. Mathematics Subject Classification...
متن کامل