Projective-metric geometry

A projective-metric geometry is an at least two-dimensional projective geometry over a body with an additional metric structure . This additional structure enables the “orthogonality relation” of a metric absolute geometry to be described in the projective space in which the metric absolute geometry can be embedded. At the same time, this additional structure characterizes a certain subgroup in the group of all projectivities of space: It is the invariance or compatibility group of polarity or polar involution , through which the orthogonality of non-Euclidean or Euclidean geometry is represented in projective space.

In two main cases of absolute geometry , elliptical geometry and Euclidean geometry , the projective space together with the “polar structure” uniquely determines the embedded geometry. Only in the case of elliptical geometry does projective space itself, with its projective polarity, represent a model of absolute geometry.

The projective-metric geometry is richer than the metric absolute geometry in the following sense : There is not an absolute geometry for every projective-metric space.

On the other hand, the projective-metric geometry is “forgetful”: For the main hyperbolic case and all secondary cases, e.g. B. the semi-elliptical planes, the original metric plane can generally only be uniquely recovered after embedding down to isomorphism if, in addition to the invariance group, which is uniquely determined by the polar structure, one also has a certain generating system of this invariance group, which in of the metric plane represents the vertical axis reflections and the straight lines of the geometry.

Definitions and basic terms in the flat case

The basic terms are shown here for the plane: Let it be a body and the projective plane above it , the associated coordinate vector space. The mentioned requirement that the characteristic of the observed body is not 2 is made in this context (often tacitly) because otherwise reflections cannot be meaningfully defined. ${\displaystyle K}$ ${\displaystyle \operatorname {char} K\neq 2}$ ${\displaystyle \mathbb {P} ^{2}(K)}$ ${\displaystyle K}$ ${\displaystyle V=K^{3}}$

Symmetrical bilinear shape and polar involution / polarity

A symmetrical bilinear form is given on the coordinate vector space . This can be in by a symmetric matrix represent: . ${\displaystyle B:V\times V\rightarrow K}$ ${\displaystyle K^{3}}$ ${\displaystyle A}$ ${\displaystyle B(y,x)=y^{T}Ax}$

1. Singular case: If the matrix has rank 2, then the radical of the symmetrical bilinear form is given by the kernel of the linear mapping , this is then a one-dimensional subspace of . ${\displaystyle A}$ ${\displaystyle V\rightarrow V;x\mapsto Ax}$ ${\displaystyle \langle x_{R}\rangle }$ ${\displaystyle V}$
2. Non-Euclidean, vulgar case: If the matrix full rank 3, and is therefore regularly , then the linear transformation describes the point image of a projective polarity on . ${\displaystyle A}$ ${\displaystyle V\rightarrow V;x\mapsto (Ax)^{T}}$ ${\displaystyle \mathbf {P} ^{2}(K)}$

For the singular case 1 and above all for the higher-dimensional generalizations of this case, it is practical to use the one-dimensional subspaces of as straight lines of the projective plane and the two-dimensional subspaces as projective points in all cases . This convention is dual to the procedure that is usually usual. Thus the radical in the singular case is a projective distance line distinguished by the bilinear form . ${\displaystyle V}$ ${\displaystyle \langle x_{R}\rangle }$ ${\displaystyle B}$ ${\displaystyle \mathbf {g} _{\infty }}$

Orthogonality and polar involution in the singular case

In the singular case, a distinction is made between two subcases:

1. All isotropic vectors are contained in the radical, formally: or ${\displaystyle x^{T}Ax=0\Rightarrow Ax=0}$
2. there are isotropic vectors that are not contained in the radical.

In the first case, the projective plane with the bilinear form is called singular-elliptical, in the second, singular-hyperbolic . Only in the first case is the plane a projective-metric plane . In this case it is called the singular projective-metric plane . ${\displaystyle V}$

In both singular subcases, the following applies: On the "tuft" of the two-dimensional subspaces of , which contain the distance line (projective: the set of points that intersects with the distance line), the bilinear form defines an involutorial orthogonality mapping through which each of these subspaces of the to it (in the sense of the bilinear form) perpendicular two-dimensional subspace is assigned. Since this assignment can be represented as a linear mapping of coordinate vectors, there is a projective polar involution on the projective long-range line. ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle V}$

One-dimensional subspaces of the vector space, which are different from the radical , each span a two-dimensional subspace with the radical. In projective space this means: projective straight lines that are different from the long-distance straight line each define a far point . Such projective straight lines are called orthogonal if these associated two-dimensional subspaces (projective: the far points of ) are mapped onto one another through polar involution. In the singular-elliptical case, no projective straight line except the long-distance straight line is perpendicular to itself. ${\displaystyle \langle a\rangle ,\langle b\rangle }$ ${\displaystyle \langle x_{R}\rangle }$ ${\displaystyle \mathbf {a} ,\mathbf {b} }$ ${\displaystyle \mathbf {g} _{\infty }}$ ${\displaystyle \mathbf {g} _{\infty }}$ ${\displaystyle \mathbf {a} ,\mathbf {b} }$ ${\displaystyle \langle a,x_{R}\rangle ,\langle b,x_{R}\rangle }$ ${\displaystyle \mathbf {a} ,\mathbf {b} }$

It is agreed: The pole of a projective straight line is the common far point of all straight lines perpendicular to it. The distance line has no pole, polar are not defined in the singular case. ${\displaystyle \mathbf {a} \neq \mathbf {g} _{\infty }}$

Polarity in the ordinary cases

A projective plane with a bilinear form , which is described by a symmetric matrix of full rank, is called an ordinary projective-metric plane . Analogous to the singular case, a distinction is made between two subcases: ${\displaystyle V}$

1. Only the zero vector is isotropic, formally: or ${\displaystyle x^{T}Ax=0\Rightarrow x=0}$
2. genuinely isotropic vectors exist.

In the first sub-case the projective-metric plane is called elliptical , in the second it is called hyperbolic . In both subcases, there is a two-dimensional subspace for each vector different from the zero vector , projectively the pole of the projective straight line represented by. ${\displaystyle x\in V}$ ${\displaystyle \langle x\rangle ^{\perp }=\{z\in V:x^{T}Az=0\}}$ ${\displaystyle \langle x\rangle }$

1. In the elliptical case there are no genuinely isotropic vectors: no projective point incises with its polar . ${\displaystyle \langle x\rangle ^{\perp }}$ ${\displaystyle \langle x\rangle }$
2. In the hyperbolic case, the following applies to the genuinely isotropic vectors : For these vectors the projective point incurs with its polar . ${\displaystyle x\in \langle x\rangle ^{\perp }}$ ${\displaystyle x\neq 0}$ ${\displaystyle \langle x\rangle ^{\perp }}$ ${\displaystyle \langle x\rangle }$

Actually-orthogonal group and axis reflections in vector space

The description of the orthogonal group and the mirror images is independent of the projective interpretation. Due to the bilinear form (which should correspond to one of the three cases described above singular-elliptical , elliptical or hyperbolic ), the vector space has an orthogonality structure on the set of its subspaces in the usual sense of linear algebra. ${\displaystyle V}$ ${\displaystyle B}$

A bijective linear mapping is called an orthogonal transformation if it is compatible with the bilinear form: The following must then apply to all : ${\displaystyle \mathbf {S} :V\rightarrow V;x\mapsto Sx}$ ${\displaystyle B}$ ${\displaystyle x,y\in V}$

${\displaystyle B(S(x),S(y))=B(x,y)}$In matrix notation: . ${\displaystyle x^{T}\left(S^{T}AS\right)y=xAy}$

The orthogonal transformations form a group, the orthogonal group . The orthogonal transformations with determinant 1 form a subgroup of index 2, the actually orthogonal group . Proportional shapes define the same orthogonal group. ${\displaystyle O_{3}(K,B)}$ ${\displaystyle O_{3}^{+}(K,B)}$ ${\displaystyle r\cdot A,r\in K^{*}}$

For a non-isotropic vector , the linear transformation that maps every vector that is too proportional to itself and every vector that is too perpendicular to its opposite vector is thus with ${\displaystyle x\in V}$ ${\displaystyle \sigma =\sigma _{\langle x\rangle }}$ ${\displaystyle x}$ ${\displaystyle x}$

${\displaystyle \sigma (u)=u}$for and for ${\displaystyle u\in \langle x\rangle \quad }$ ${\displaystyle \quad \sigma (u)=-u}$ ${\displaystyle u\in \langle x\rangle ^{\perp }}$

an involutor orthogonal transformation. It can be represented using the bilinear form as

${\displaystyle \sigma (u)=-u+2\cdot {\frac {B(u,x)}{B(x,x)}}\cdot x}$.

It is called the mirror image of . ${\displaystyle \langle x\rangle }$

The group is generated from the mirror images . To display any group element, a maximum of three axis reflections are required. ${\displaystyle O_{3}^{+}(K,B)}$ ${\displaystyle \sigma _{\langle x\rangle },B(x,x)\neq 0}$

Overview of the related terms of the projective-metric plane and the vector space with bilinear form : ${\displaystyle K^{3}}$ ${\displaystyle B}$

Projective-metric plane	Vector space with bilinear shape
Straight ${\displaystyle \mathbf {a} }$	One-dimensional subspace ${\displaystyle \langle a\rangle }$
Point ${\displaystyle P}$	Two-dimensional subspace with the homogeneous equation ${\displaystyle U(p^{T})}$ ${\displaystyle p^{T}}$
${\displaystyle P}$, incidental ${\displaystyle \mathbf {a} }$	${\displaystyle \langle a\rangle <U(p^{T})}$
${\displaystyle \mathbf {a} ,\mathbf {b} }$ orthogonal	${\displaystyle b\in \langle a\rangle ^{\perp }}$ this means ${\displaystyle B(a,b)=0}$
${\displaystyle \mathbf {a} }$ self-orthogonal / isotropic	${\displaystyle a}$ isotropic that is ${\displaystyle B(a,a)=0}$
Reflection (involutor perspective) with the non-self-orthogonal straight line as the axis and the pole of as the center ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {a} }$	Mirroring , for ${\displaystyle \sigma _{\langle a\rangle }}$ ${\displaystyle B(a,a)\neq 0}$

Projective description of the orthogonal groups and axis reflections in the plane case

In all three cases, the ordinary-elliptical, ordinary-hyperbolic and singular-elliptical case, the bilinear form for each projective non-isotropic straight line defines a definite pole that does not intersect with the straight line. It is defined: The axis reflection on the non-isotropic straight line is the clearly determined involutive perspectivity , with the straight line as the axis and its pole as the center. According to construction, this is a projectivity. ${\displaystyle \mathbf {a} }$ ${\displaystyle \sigma _{\mathbf {a} }}$ ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathbf {a} }$

The set of all axis reflections defined in this way on non-isotropic straight lines creates a group that is isomorphic to the actually orthogonal group of the respective case defined above in the vector space. ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}=\langle {\mathfrak {S}}\rangle }$

Projective-metric planes in absolute geometry

→ Main article : Metric absolute geometry

Every metric plane in the sense of metric absolute geometry defines a projective-metric ideal plane in which it can be embedded. This projective-metric plane becomes unambiguous in that only actual straight lines pass through an actual point of the ideal plane (i.e. a point that was already present in the metric plane) . In other words, the metric plane is a locally complete sub-plane of the projective-metric plane. As a result, the projective-metric plane fulfills two tasks in absolute geometry:

1. Projective planes contain geometrically the metric planes as locally complete partial planes with the symmetric bilinear forms definable via their coordinate vector space. The orthogonality of such a sub-plane, as a relation between its straight lines, is clearly determined by the ordinary or singular-elliptical polar structure, which determines the bilinear shape on the projective plane, i.e. its “metric additional structure”.
2. The orthogonal group of a projective-metric plane containing group theoretically the movement groups axiomatic defined as generated subgroups and thus provides a representation of these groups. ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$

Different metric levels can have the same ideal level. The following assignment exists between the axiomatically described movement groups and their projective-metric ideal level:

Additional axioms of the metric plane	designation	Ideal level
Axioms and ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {V} ^{*}}$	Pre-Euclidean level	(singular) projective-Euclidean plane
Axioms and ${\displaystyle \mathbf {R} }$ ${\displaystyle \neg \mathbf {V} ^{*}}$	Metric-Euclidean plane with no clear parallel
axiom ${\displaystyle \mathbf {P} }$	elliptical plane	(Ordinary) projective-elliptical plane
Axioms ${\displaystyle \neg \mathbf {R} ,\neg \mathbf {P} ,\mathbf {V} ^{*}}$	semi-elliptical plane
Axioms and ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle \mathbf {H} }$	hyperbolic plane	(Ordinary) projective-hyperbolic plane

If one tries, starting from a projective-metric plane, to construct a metric plane as a group generated from reflections , then one has the actually orthogonal group as a group and an invariant generating system of this group has to be defined as a set of axis reflections so that the generated Group fulfills the desired axioms. In the elliptical case one can use all reflections on projective straight lines, in the Euclidean case all reflections on non-isotropic straight lines (these are all projective straight lines except one, which becomes the long-distance line of the metric plane). You then have the corresponding main case highlighted in the table as the metric level. ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$ ${\displaystyle {\mathfrak {G}}\;}$ ${\displaystyle {\mathfrak {S}}\;}$ ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$

In the hyperbolic case, a class of actual straight lines and the resulting axis reflections can only be identified if the body is arranged or allows an arrangement, i.e. is formally real. ${\displaystyle K}$

In the main hyperbolic case of absolute geometry, the existence of the projective-metric plane (i.e. a hyperbolic projective polarity on a projective plane) is not sufficient for the existence of a corresponding metric plane. The actually-orthogonal groups of the projective-hyperbolic planes are called H-groups in absolute geometry and can be described axiomatically and abstractly in a similar way as the metrically generated movement groups . ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}}\;)}$

Generalization and language regulations for higher dimensions

In an n -dimensional projective space over a body , the characteristic of which is not two, one can explain a polar structure analogously to the plane case using a bilinear shape . Here again in the singular-elliptic case the one-dimensional radical becomes a remote hyperplane, and in all cases a pole (the well-defined hyperplane of the vector space that may contain the radical) can be used for the product of non-isotropic vectors x (hyperplanes of projective space ) To be defined. In place of the axis reflections, there are then hyperplane reflections or axial collineations, etc. ${\displaystyle K,(n>2)}$ ${\displaystyle V=K^{n+1}}$ ${\displaystyle \langle x\rangle ^{\perp }}$

Here, too, a distinction is made between the three cases mentioned above. The following table gives an overview of the designation for the respective projective-metric geometry , whereby the singular-hyperbolic case is also included, which is not examined in the absolute geometry (in the narrower sense).

	Every isotropic vector is contained in the radical.	There are isotropic vectors outside the radical.
The radical is the null space.	Ordinary-elliptical, briefly elliptical	ordinary hyperbolic, hyperbolic for short
The radical is one-dimensional.	singular-elliptical, Euclidean for short	singular hyperbolic, Minkovskian for short

The respective short attribute can then replace metric , for example a three-dimensional projective-metric geometry, with an elliptical bilinear form of rank 4, three-dimensional projective-elliptic geometry or a three-dimensional projective-elliptic space and a 5-dimensional projective-Euclidean space is a 5-dimensional projective space with a polar structure, which is determined by a zero-part bilinear form of rank 5, etc.

literature

Textbook (main source)

• Friedrich Bachmann: Structure of geometry from the concept of reflection . 2nd supplemented edition. Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 .

References and comments

1. ↑ ^{a b} Apart from a few special cases in which the movement group of the metric level is itself a real subgroup of the actually orthogonal group, Bachmann (1973) §18 On the metric movement groups . ${\displaystyle {\mathfrak {G}}}$
2. ↑ The formalities of linear algebra - coordinate vectors are shown here as column and plane coordinates of two-dimensional subspaces as row vectors and are objects of the dual space - are not of central importance here, but should be taken into account in order to avoid misunderstandings. Bachmann (1973) uses the index notation for coordination, without matrices. This is just as dependent on coordinates as the matrix notation, Bachmann (1973), §8.3 Projective-metric coordinate planes and metric vector spaces. ${\displaystyle V^{*}}$
3. ↑ “Practical” but mathematically not absolutely necessary, Bachmann (1973), § 8.3
4. ↑ Bachmann (1973) §8.3
5. ↑ Bachmann (1973), p. 90. In the literature, far points that are assigned to one another through polar involution are also called polar , in line with the fact that the projective polar involution can be viewed as a projective polarity on the long line.
6. ↑ Bachmann (1973), §8.4 Projective-metric coordinate planes and metric vector spaces.
7. ↑ Bachmann (1973), §9.1, sentence 2
8. ↑ Bachmann (1973), §9.1 Projective-metric geometry theorems 4, 5 and 6
9. ↑ according to Bachmann (1973) §20.7 Supplement