Affine semipolar spaces

Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an affine semipolar space arises as a result of generalization of the notion of an alternating form to a semiform. Some properties of these two geometries are given and their automorphism groups are characterized.


Introduction
Dealing with symplectic affine polar spaces we observe some regularities that lead to a new notion: semiform. In turn semiforms give rise to an interesting class of quite general partial linear spaces called affine semipolar spaces.
In [5] an affine polar space (APS in short) is derived from a polar space the same way as an affine space is derived from a projective space, i.e. by deleting a hyperplane from a polar space embedded into a projective space. Such affine polar spaces are embeddable in affine spaces and this let us think of them as of suitable reducts of affine spaces.
In general we have two types of APS'es. Structures of the first type are associated with polar spaces determined by sesquilinear forms; one can loosely say: these are "stereographical projections of quadrics". They can also be thought of as determined by sesquilinear forms defined on vector spaces which represent respective affine spaces. These structures and adjacency of their subspaces were studied in [17]. Contrary to [5], in this approach Minkowskian geometry is not excluded. In particular, the result of [17] generalizes Alexandrov-Zeeman theorems originally concerning adjacency of points of an affine polar space (cf. [1,19]).
The second class of APS'es, which is included in [5] but is excluded from [17], consists of structures associated with polar spaces determined by symplectic polarities. The aim of this paper is to present in some detail the geometry of the structures in this class from view of the affine space in which they are embedded. The position of the class of thus obtained structures-let us call them symplectic affine polar spaces-is in many points a particular one.
Firstly, symplectic affine polar spaces are associated with null-systems, quite well known polarities in projective spaces with all points selfconjugate. So, symplectic APS'es have famous parents. Moreover, in each even-dimensional pappian projective space such a (projectively unique) polarity exists. Thus a symplectic APS is not an exceptional space, but conversely, it is also a "canonical" one in each admissible dimension.
A second argument refers to the position of the class of symplectic APS'es in the class of all APS'es. As an affine polar space is obtained by deleting a hyperplane from a polar space, while the latter is realized as a quadric in a metric projective space, the derived APS appears as a fragment of the derived affine space. If the underlying form that determines the polar space is symmetric then the corresponding APS can be realized on an affine space in one case only-when the deleted hyperplane is a tangent one. And then the affine space in question is constructed not as a reduct of the surrounding projective space but as a derived space as it is done in the context of chain geometry (cf. [2,8]). Moreover, such an APS can be represented without the whole machinery of polar spaces: it is the structure of isotropic lines of a metric affine space.
The only case when the point set of the reduct of a polar space is the point set of an affine space arises when we start from a null system i.e. in the case considered in the paper. But such an APS is not associated with a metric affine space i.e with a vector space endowed with a nondegenerate bilinear symmetric form. What is a natural analytic way in which a symplectic APS can be represented, when its point set is represented via a vector space? A way to do so is proposed in our paper: to this aim we consider a "metric", a binary scalar-valued operation defined on vectors. It is not a metric, in particular, it is not symmetric, and it is not invariant under affine translations. Nevertheless, it suffices to characterize respective geometry.
Symplectic APS'es have famous parents but they have also remarkable relatives. Although the "metric": the analytical characteristic invariant of symplectic APS'es is not a form, it is closely related to forms. Loosely speaking, it is a sum of an alternating form η defined on a subspace and an affine vector atlas defined on a vector complement of the domain of η. Immediate generalization with 'an alternating map' substituted in place of 'an alternating form' comes to mind. Such a definition of a map may seem artificial. The resulting map, that we call a semiform, has quite nice synthetic characterization though. Their basic properties are established in Sect. 2. A symplectic "metric" appears to be merely a special instance of such a general definition and many problems concerning it (so as to mention a characterization of the automorphism group) can be solved in this general setting easier. To illustrate and to motivate such a general definition we show in Example 2.5 a semiform associated with a vector product, that yields also an interesting geometry. On the other hand, this geometry has close connections (see Example 2.5-B ) with a class of hyperbolic polar spaces.
A semiform induces an incidence geometry that we call an affine semipolar space [cf. (11) and (12)]. It is a Γ-space with affine spaces as its singular subspaces (cf. Theorem 3.5), and with generalized null-systems comprised by lines and planes through a fixed point (cf. Proposition 3.13; comp. a class with similar properties considered in [6]). In the paper we do not go any deeper into details of neither geometry of semiforms nor geometries other than symplectic APS'es. We rather concentrate on "APS'es and around".
Finally, we pass to our third group of arguments: that geometry of symplectic APS'es is interesting on its own right. Geometry of affine polar spaces is, by definition, an incidence geometry i.e. an APS is (as it was defined both in [5] and [17]) a partial linear space: a structure with points and lines. From the results of [4] we get that geometry of symplectic affine polar spaces can be also formulated in terms of binary collinearity of points-an analogue of the Alexandrov-Zeeman Theorem. A characterization of APS'es as suitable graphs is not known, though.
The affine polar spaces associated with metric affine spaces (as it was sketched above) can be, in a natural consequence, characterized in the "metric" language of line orthogonality or equidistance relation inherited from the underlying metric affine structure. It is impossible to investigate a line orthogonality imposed on an affine structure so as it gives rise to a symplectic APS. However, in case of a symplectic APS a "metric" mentioned above determines an "equidistance" relation which can be used as a primitive notion to characterize the geometry.

Definitions and preliminary results
Let S be a nonempty set, whose elements will be called points, and let L be a family of at least two element subsets of S, whose elements will be called lines. A point-line structure M = S, L is said to be a partial linear space whenever two of its distinct points lie in at most one line. Two points a, b ∈ S are collinear if there is a line l ∈ L such that a, b ∈ l; then we write a, b = l. We call M a Γ-space if it satisfies a so called none-one-or-all axiom stating that for all a ∈ S and l ∈ L the point a is collinear with none, one or all of the points of the line l. A sequence of lines l 0 , l 1 , . . . , l n is called a path in M if l i meets l i−1 in some point for all i = 1, 2, . . . n. We say that two points a, b are joinable in M if there is a path l 0 , l 1 , . . . , l n such that a ∈ l 0 and b ∈ l n . If every two points in M are joinable we call M connected.
Recall that the affine space A(V) defined over a vector space V has the vectors of V as its points and the cosets of the 1-dimensional subspaces of V as its lines.

Polar spaces
Let W be a vector space over a (commutative) field F with characteristic = 2 and let ξ be a nondegenerate bilinear reflexive form defined on W. Assume that the form ξ has finite index m and n = dim(W). We will write Sub(W) for the class of all vector subspaces of W and Sub k (W) for the class of all k-dimensional subspaces. In the projective space P = Sub 1 (W), Sub 2 (W), ⊂ the form ξ determines the polarity δ = δ ξ . We write Q(ξ) for the class of isotropic subspaces of W: Assume that m ≥ 2. The structure is referred to as the polar space determined by δ in P. Note that k-dimensional isotropic subspaces of W are (k−1)-dimensional singular subspaces of the polar space Q ξ (W).

Hyperbolic polar spaces and their reducts
This section may look superfluous from view of symplectic polar spaces but it is used later in Example 2.5-B which justifies our general construction of semiforms.
Assume that the form ξ on W is symmetric and set Y := W ×W , Y := W⊕W, Z := W × {θ} (θ being the zero vector), is symmetric, nondegenerate, and Sub 1 (H) = Q 1 (ζ). Since Z is a n-dimensional maximal isotropic subspace of 2n-dimensional vector space Y equipped with the form ζ, the projective index of Q := Q ζ (Y ) is n − 1. Hence Q is a hyperbolic polar space (or a hyperbolic quadric, cf. [15, Sec. 1.3.4, p. 30]). Now let Z be a maximal, i.e. (n − 1)-dimensional, singular subspace of a hyperbolic polar space Q of projective index n − 1 and let R = R(Q, Z) be the structure obtained by deleting the subspace Z from Q. We also write R(W, ξ) = R(Q ζ (Y ), Sub 1 (Z)).
Since Z is contained in a hyperplane of Q the next theorem follows from [16,Theorem 3.11] which says that the ambient, thick, nondegenerate, embeddable polar space of rank at least 3 can be recovered in the complement of its subspace that is contained in a hyperplane. We give an independent, not so complex proof based on the decomposition of the hyperbolic polar space Q.
Proof. We need to recover in R points and lines of Z which are missing to get Q. Let C be the family of all maximal singular subspaces of Q and let R be the family of all maximal singular subspaces of R. It is seen that R = {X \Z : X ∈ C}, and thus, every element of R carries the geometry of a slit space (cf. [11,12]). Write -R 0 = {X \Z : dim(X ∩ Z) = 0} for the family of punctured projective spaces, every one of which determines on Z its improper point, -R 1 = {X \Z : dim(X ∩ Z) = n − 2} for the family of affine spaces, every one of which determines on Z a (n − 2)-subspace of its improper points.
As Q is a dual polar space of type D n , in view of [14,Sec. 4.2.3] the family C can be uniquely decomposed into the disjoint union of two subsets C + and C − (the half-spaces) such that Correspondingly, the family R can be decomposed into We assume that Z ∈ C − . Then Taking into account that J 0 has a unique improper point and using the decomposition into half-spaces which gives that 1 ≤ dim(X 0 ∩ X 1 ) independently on n we have that whenever Note that is an equivalence relation and J 0 J 0 means that J 0 , J 0 share the improper point. Therefore, there is a one-to-one correspondence between the equivalence classes of the relation and the points of Z.
Next, for every (n−2)-dimensional singular subspace N of Q there are precisely two maximal singular subspaces containing N , one of them is from C + and the other belongs to C − . This gives a one-to-one correspondence between the elements of R 1 and the hyperplanes of Z.
That way, in terms of R, we get an incidence structure with points and hyperplanes of Z. Using standard methods we are able to recover lines of Z which makes the proof complete for Z ∈ C − . In case Z ∈ C + the reasoning runs the same way.
Let H 0 be a hyperplane of Q (cf. [5]); then H 0 is determined by a hyperplane H of P; on the other hand H is a polar hyperplane of a point U of P i.e. H = U ⊥ . Finally, H 0 = H is the set of all the points that are collinear in Q with the point U of Q. The affine polar space U derived from (Q,U ) is the restriction of Q to the complement of H; in view of the above the point set of U is the point set of the affine space A obtained from P by deleting its hyperplane H. The set G of all the lines of U is a subset of the set L of the lines of A. Moreover, the parallelism of the lines in G defined as in [5] (two lines are parallel iff they intersect in H 0 ) coincides with the parallelism of A restricted to G. Clearly, not all the lines of P that are not contained in H are isotropic. Furthermore, none of the lines of P through U which is not contained in H is isotropic. For this reason, in every direction of A, except the one determined by U , there is a pair of parallel lines in A such that one of them is isotropic and the other is not. In this exceptional direction no line is isotropic.
In [17] affine polar spaces determined in metric affine spaces associated with symmetric forms were studied. A somewhat similar interpretation of U can be given here as well.
Recall that there is a basis of W in which the form ξ is given by the formula Let us take U = [0, 1, 0, . . . , 0] ; then H is characterized by the condition [x 1 , . . . , x n ] ⊂ H iff x 1 = 0. We write V for the subspace of W characterized by x 1 = x 2 = 0; note that the restriction η of ξ to V is also a nondegenerate symplectic form. We can write W = F ⊕ F ⊕ V and then for scalars a 1 , a 2 , b 1 , b 2 and vectors u 1 , with scalars a 1 , a 2 and u 1 , u 2 ∈ V be a pair of points of A. Then Proof. Embed the points p 1 , p 2 into P; then p i corresponds to Since p 1 , p 2 are collinear iff the projective line which joins U 1 , U 2 is in Q we get that p 1 , p 2 are collinear iff ξ(U 1 , U 2 ) = 0. By (1) we get our claim.
The resulting map is referred to as a semiform defined on Y.
If η is a bilinear map, then we write η u for the map defined by η u (v) = η(u, v). An alternating bilinear form η, like the one considered in Definition 2.1, is The formulas that are coming next are technical but quite important. They follow immediately from the definition.
What follows are various examples of more or less natural semiforms.  (1).
(see any standard textbook, e.g. [13, Ch. XIX]). It is known that dim( If we take g = id, then we get a specific bilinear alternating map η with the property that dim(ker(η u )) = 1 for all nonzero u ∈ V as for linearly independent u 1 , u 2 the wedge product u 1 ∧ u 2 cannot be zero.
Applying (2), the map η together with some suitable δ gives rise to a semiform.
In case dim(V i ) = n < ∞, for some i, it is possible to decompose η i into n alternating bilinear formsη j i : V × V −→ F , where F is the ground field of V, V i and j = 1, . . . , n so that u 2 ), . . . ,η n i (u 1 , u 2 ) . If dim(V ) < ∞ we can do the same with η.

Affine atlas and its characterization
In this and the forthcoming Sects. 2.3 and 2.4 most of the proofs consist in direct computations and therefore they are left for the reader.
Let us give a more explicit representation of a map δ characterized in Definition 2.1(ii).  δ(v, 0). Then φ is a linear map and δ is characterized by the formula A map δ defined by formula (9) is called an affine atlas. It is said to be nondegenerate when φ is an injection (i.e. if ker(φ) is trivial). Note that when dim(V ) < ∞ and δ is nondegenerate then the representing map φ is a surjection as well.
Finally, we note that affine atlases can be equivalently characterized by another, less elegant but more convenient for our further characterizations, set of postulates.

Clearly, M = ker( θ ) and thus M is a subspace of Y.
If, moreover, Axiom A4 is valid then the assignment M p −→ p is injective.

Then θ ∈ D and the set D is closed under vector addition.
(ii) Assume that Axiom A6 is valid. Then q ∈ D iff (p 1 + q, p 2 + q) = (p 1 , p 2 ) for all p 1 , p 2 ∈ Y . (iii) Set

If Axiom A7 is valid, then the set D is closed under scalar multiplication.
If Axiom A5 is adopted, then D = D .   16. Let dim(V) = 2. Then the determinant is a symplectic form. Therefore the following map is a semiform (x i , y i are scalars).

A simplification of semiforms
Forthcoming constructions are provided for a fixed nondegenerate semiform defined in Definition 2.1. Moreover, we assume that dim(V ) =: ν < ∞.
for suitable maps η, φ. Recall that they need to be nondegenerate. As a consequence, φ ∈ GL(V ).
There is, generally, a great variety of semiforms. But some of them may lead to isomorphic geometries. Write η,φ for defined by Definition 2.1 with δ defined by (9). We have evident Proposition 2.17. (i) There is a linear bijection Φ ∈ GL(Y) such that for any q 1 , q 2 ∈ Y it holds: (ii) Let B ∈ GL(V), γ be a nonzero scalar. Then, clearly, the map γηB defined by γηB(u 1 , u 2 ) = γ·η(B(u 1 ), B(u 2 )) is an alternating form. There is a linear bijection Φ ∈ GL(Y) such that the following holds for any Remark. In terms of Example 2.3 we have ηB = g • (B ∧ B).
In view of Proposition 2.17, till the end of our paper we assume that is defined by a formula of the form

Geometrical structure
Under the settings of Sect. 2.4 that A = A(Y) and p 1 , p 2 are points of A with imitating Lemma 1.2, we put generally From definition it is immediate that Lemma 3.1. Let p 1 , p 2 be two distinct points of A and L = p 1 , p 2 . If p 1 ∼ ∼ p 2 , then q 1 ∼ ∼ q 2 for all q 1 , q 2 ∈ L.
In view of Lemma 3.1, the relation ∼ ∼ determines the class G of lines of A by the condition For computation purposes it is convenient to have this criteria: It is a straightforward consequence of (12) and (11).
The class G induces the partial linear space that we call affine semipolar space determined by . Let us put down some of its properties.

Lemma 3.3. (i)
The class G is not closed under parallelism, i.e. for every This yields that η(u, u 0 ) = v for all u 0 ∈ V . So take any u 1 ∈ V and note that η(u, Thus u ⊥ = V , and hence u = θ as η is nondegenerate. Finally, [v, u] is the zero of Y, a contradiction. (ii): Without loss of generality we can assume that p = [v 0 , u 0 ] and L i = p+ a i where a i ∈ Y , i = 1, 2. Then L = p + α 1 a 1 + α 2 a 2 for some α i ∈ F . Applying (13) to L 1 , L 2 and then to L we are through.
Note that Lemma 3.3(ii) means that the family G is closed on pencils. This has some straightforward implication.

Corollary 3.4. The set of points that are collinear with a given point in an affine semipolar space (determined by a semiform defined on Y) is closed on planes and thus it is a subspace in the ambient affine space A.
Theorem 3.5. The affine semipolar space is a Γ-space and its every singular subspace carries affine geometry.
Proof. Let U be our affine semipolar space. The first part follows directly from (ii) in Lemma 3.3. The other part is a simple observation that a singular subspace of U, in other words, a strong subspace wrt. ∼ ∼ in A or a subspace where every two points are ∼ ∼ -adjacent, is an affine subspace of A.
When we deal with a Γ-space a question on the form of its triangles may appear important. The following is immediate from (13) and (11).
Proposition 3.6. If dim(ker(η u )) = 1 for each nonzero u ∈ V , then the corresponding affine semipolar space contains no proper triangle. In that case its maximal singular subspaces are the lines.
Example 3.7. In view of Theorem 3.5 one could expect that affine semipolar spaces are models of the system considered in [6]. In the case of wedge product considered in Example 2.3 however, and consequently in the case of vector product considered in Example 2.5, we have dim(ker(η u )) = 1 for all u = θ. By Remark 3.2 it is seen that there are no triangles in the affine semipolar space determined by η with this property. Therefore, affine semipolar spaces from Example 2.3 and Example 2.5 are not models of the system in [6].
Example-continuation 2.4-A Assume additionally that V 0 = · · · = V ν and η 0 is a linear combination of the other η i , that is for some scalars λ i , i = 1, . . . , ν. By (11), in the affine semipolar space . This means that U is not connected.
In the sequel we shall frequently consider the condition (with prescribed values u, v) applying the representation given in Example 2.3 this can be read as (∃y) [ g(u∧ . This observation allows us to construct quite "strange" ('locally surjective') alternating maps.
As an immediate consequence of Lemma 3.1 and the definition we have (i) There is no line L ∈ G with the direction q.
(ii) The equation is not solvable in u.
In particular, if u 0 = θ and v 0 = 0 then (15) is not solvable and thus there is no line L ∈ G with the direction q. Example-continuation 2.5-A Let × be a vector product in a vector 3-space V associated with a nondegenerate bilinear symmetric form ξ and ⊥=⊥ ξ be the orthogonality determined by ξ. Then for u 0 , v 0 = θ Eq. (15) is solvable iff u 0 ⊥ v 0 . In that case we have: Lemma 3.10. For a fixed u 0 ∈ V , v 0 ∈ V and a scalar α the set is a subspace of A. The class of sets of form (17) is invariant under translations of A.
Proof. Take [v 1 , u 1 ], [v 2 , u 2 ] ∈ Z and an arbitrary scalar λ. Then we compute This proves that Z is a subspace of A.
For α = −1 in (17), directly by (11), the set Z is the set of all points collinear with [v 0 , u 0 ] in U. Hence, applying Lemma 3.11 we get that the subspace in Corollary 3.4 has dimension dim(V).
Hence w ∈ ker(η y ) : y ∈ ker(η u ) . Applying the global assumptions we infer w u and thus x ∈ L. Now, we are going to make a few comments that together with Remark 3.2 will let us characterize the geometry of the lines and the planes through a point in an affine semipolar space.
Each alternating map η : V × V −→ V determines the incidence substructure Q η (V) of the projective space P(V) with the point set unchanged and with the class L * of projective lines of the form u 1 , u 2 , where u 1 , u 2 ∈ V are linearly independent and η(u 1 , u 2 ) = 0 as its lines. With a fixed basis of V one can write η as the (Cartesian) product of ν bilinear alternating forms where ν = dim(V ). Clearly, the η i need not to be nondegenerate. So, each η i determines a (possibly degenerate) null system Q ηi (V) with the lines Q 2 (η i ). The class L * is simply

Automorphisms
Recall that the horizon of an affine space, in other words, the set of all the points at infinity, can be endowed with an incidence structure and as such carries projective geometry. To establish the automorphism group of the relation ∼ ∼ and U we need some additional assumption that the set of directions V × {θ} can be characterized in terms of the projective geometry of the horizon of A(Y) with the set D [defined in (16)] distinguished. It is hard to give a formal, precise formula stating that. Let us put it this way, in the language of automorphisms.
(**) Every automorphism of the projective geometry of the horizon of A(Y) that leaves invariant the set D defined in (16)   Recall by (12) that an affine line is in G iff any two of its distinct points are ∼ ∼ related. Consequently, if F is an affine transformation of A, then F preserves G iff F preserves ∼ ∼ .
In that case the semiform is transformed under the rule for p 1 , p 2 ∈ Y . In consequence, F is an affine (linear) automorphism of U.
Conversely, under additional assumption that (**) is valid, each affine automorphism of U is of the form (18).
Note. If η is onto V then given map ϕ, condition (b) uniquely determines ψ 1 . Similarly, for a given map ϕ and vector u 0 , condition (a) uniquely determines ψ 2 .
Example-continuation 2.4-B If η i , for some i ∈ {0, . . . ν}, is a linear combination of the other η i , then, as U is not connected, automorphisms of U need not to be given by linear or semilinear maps. According to Proposition 3.16 affine semipolar space is homogeneous which together with Proposition 3.13 gives the following: Consequently, the class of lines of U is definable in terms of the binary collinearity ∼ ∼ of U.
Proof. In view of Proposition 3.16 without loss of generality we can assume that p = [0, θ] and then Lemma 3.12 yields the claim directly.