Euclidean geometry of orthogonality of subspaces

The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Prażmowska et al. (Linear Algebra Appl 430:3066-3079, 2009) and Prażmowska andŻynel (Adv Geom, to appear). Mathematics Subject Classification (2010). 51F15, 51A45.A quadric Q in a projective space (or, in a more "abstract" and general language, a polar space embedded into a projective space) may determine quite various incidence geometries. One can consider the geometry on Q; the points and lines of this geometry are projective points and lines which lie on Q. This is a primary model of polar geometry. One can take Q and the lines which are intersections of Q and secants of Q (cf.[10]). One can consider as points the projective points not on Q, and as the lines projective lines which are tangent to Q (cf. [9]) or which are projective nonsingular (nonisotropic) or hyperbolic (cf. [12,13]).In this huge variety of incidence systems related to quadrics a very important subspaces are those radical free, called regular subspaces. Motivations to consider them come from reflection geometry, a quite natural language in which a geometry is characterized in terms of its admissible reflections (cf. [2,3,20]). Axes of such reflections are exactly those subspaces which we call regular. The concept of regular subspaces can be adopted in metric projective as well as in metric affine geometry and this is what we are doing here.

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“…[17]) to orthogonality of arbitrary subspaces, and in investigations on reflections in subspaces in our metric spaces (cf. [1,22]).…”

Section: Formulation Of Problemsmentioning

confidence: 99%

“…And we want our interpretations to keep, in a sense, this meta-interpretation. In the paper [17] we have explained in some detail what this kind of interpretation means (see also [14,Sec. 2]).…”

Section: Axiomatic Aspectsmentioning

confidence: 99%

Grassmann spaces of regular subspaces

2010

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“…In [18] ortho-adjacency is treated more generally as a relation on all k-subspaces, not only on lines, of an Euclidean space. It is proved there that ortho-adjacency-preserving transformations on k-subspaces are induced by orthogonality-preserving collineations of the underlying n-dimensional Euclidean space where k + 2 ≤ n. In other words orthoadjacency on k-subspaces can be used as a single primitive notion for at least (k + 2)-dimensional Euclidean geometry.…”

Section: Introductionmentioning

confidence: 99%

“…The idea of ortho-adjacency relation is taken from [18] and [8]. Given a linear space with an orthogonality relation defined on its line-set, two lines are called ortho-adjacent if they are concurrent and orthogonal.…”

Section: Introductionmentioning

confidence: 99%

Orthogonality of subspaces in metric-projective geometry

Prażmowski

Żynel

2011

Advg

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“…In essence, these results state that a mapping that preserves a certain geometric notion ν must be a certain kind of geometric transformation, and thus that it needs to surprisingly preserve much more than just ν. Such characterizations have been shown in a series of papers [12][13][14][15][16][17][18][19][21][22][23] to be equivalent to definability statements which state that certain geometric notions are definable, with certain syntactic constraints on the definiens, in terms of another geometric notion. Characterizations of self-mappings of hyperbolic spaces, going beyond those that have been collected in [2][3][4], can be found in [7][8][9][10][11]13,[15][16][17]21].…”

Section: Introductionmentioning

confidence: 99%

Mappings preserving the area equality of hyperbolic triangles are motions

Pambuccian

2010

Arch. Math.

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It is shown that a mapping ϕ : A → B between models A and B of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto ϕ(A). The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for "the triangles abc and abd have the same area." Mathematics Subject Classification (2000). Primary 51M10; Secondary 51M25, 03C40.

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Euclidean geometry of orthogonality of subspaces

Abstract: We prove that Euclidean geometry is interpretable in terms of orthogonality of its k-subspaces, and thus it can be formalized as a theory of such an orthogonality. (2000). 51F20, 51M04. Mathematics Subject Classification

Cited by 10 publications

References 19 publications

Grassmann spaces of regular subspaces

Grassmann spaces of regular subspaces

Orthogonality of subspaces in metric-projective geometry

Mappings preserving the area equality of hyperbolic triangles are motions

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