Isometric embeddings of polar Grassmannians and metric characterizations of their apartments

We describe isometric embeddings of polar Grassmann graphs formed by non-maximal singular subspaces. In almost all cases, they are induced by collinearity-preserving injections of polar spaces. As a simple consequence of this result, we get a metric characterization of apartments in polar Grassmannians.

maximal simplices in the building. In this case, we say that the line joining a and b is the set of all vertices c for which P ∪ {c} is a chamber. So, every Grassmannian can be considered as a graph as well as a point-line geometry.
Every building of type A n , n ≥ 3 is the flag complex of a certain (n+1)-dimensional vector space over a division ring, and the corresponding Grassmannians are formed by subspaces of the same dimension. Similarly, every building of type C n is the flag complex of a rank n polar space and all buildings of type D n can be obtained from polar spaces of type D n . The Grassmannians of such buildings are polar and half-spin Grassmannians. The associated graphs are said to be polar and half-spin Grassmann graphs. Note that polar Grassmann graphs formed by maximal singular subspaces are known as dual polar graphs. Classical Chow's theorems [3] (see also [9,13]) describe automorphisms of dual polar graphs and half-spin Grassmann graphs. The description of automorphisms of polar Grassmann graphs formed by non-maximal singular subspaces is given [13,Section 4.6]. In almost all cases, such automorphisms are induced by collineations of the associated polar spaces (except the case of polar and half-spin Grassmann graphs of type D 4 ).
In this paper isometric embeddings of polar Grassmann graphs will be investigated. Since graph isomorphisms can be characterized as surjective isometric embeddings, the results concerning automorphisms easily follow from the description of isometric embeddings.
Let be a polar space of rank n. Denote by k ( ) the polar Grassmann graph formed by k-dimensional singular subspaces of . Let also be a polar space of rank n . By [14,Theorem 3], every isometric embedding of the dual polar graph n−1 ( ) in the dual polar graph n −1 ( ) is induced by a collinearity-preserving injection of to the quotient polar space of by a certain (n − n − 1)-dimensional singular subspace. It follows from [14,Theorem 2] that apartments in the polar Grassmannian formed by maximal singular subspaces of can be characterized as the images of isometric embeddings of the n-dimensional hypercube graph H n in n−1 ( ). If and are polar spaces of types D n and D n (respectively) and n is even, then the same holds for isometric embeddings of the associated half-spin Grassmann graphs [16,Theorem 4]. By [16,Theorem 2], apartments in the half-spin Grassmannians of can be characterized as the images of isometric embeddings of the half-cube graph 1 2 H n in the corresponding half-spin Grassmann graphs; as above, we assume that n is even. Also, there is the following conjecture [16,Section 6]: if n is odd, then there exist isometric embeddings of 1 2 H n in the half-spin Grassmann graphs of whose images are not apartments.
In this paper similar results will be established for isometric embeddings of polar Grassmann graphs formed by non-maximal singular subspaces (Theorems 1-3). Our arguments are different from the arguments given in [14,16]. In dual polar graphs and half-spin Grassmann graphs the distance between two vertices is completely defined by the dimension of the intersection of the corresponding maximal singular subspaces. For polar Grassmann graphs formed by non-maximal singular subspaces the distance formula is more complicated (Sect. 2.4).
As a simple consequence of the main results, we get the following metric characterization of apartments in polar Grassmannians (Corollary 1): if k (n) denotes the restriction of the graph k ( ) to any apartment, then the image of every isometric embedding of k (n) in k ( ) is an apartment.
It must be pointed out that there is no similar characterization for apartments in Grassmannians of vector spaces. Let V be an n-dimensional vector space (over a division ring). Consider the Grassmann graph k (V ) formed by k-dimensional subspaces of V . The restriction of k (V ) to every apartment of the corresponding Grassmannian is isomorphic to the Johnson graph J (n, k). The image of every isometric embedding of J (n, k) in k (V ) is an apartment only in the case when n = 2k. The images of all possible isometric embeddings of Johnson graphs in Grassmann graphs are described in [17,Chapter 4]. Also, [17,Chapter 3] contains the complete description of isometric embeddings of Grassmann graphs. They are defined by semilinear embeddings of special type; such semilinear embeddings are not necessarily strong.
Other characterizations of apartments in building Grassmannians can be found in [4,7,[10][11][12]15]. Most of them are in terms of independent subsets of point-line geometries.
A building Grassmannian can be contained in other building Grassmannian as a subspace (in the sense of point-line geometry). There is a natural question: is it possible to determine all such subspaces? This problem is closely related to characterizing apartments and solved for some special cases [1,[4][5][6]. For example, subspaces of polar Grassmannians isomorphic to Grassmannians of vector spaces are described in [1]. There is a similar description for subspaces of symplectic Grassmannians isomorphic to other symplectic Grassmannians [6].

Graphs
We will consider connected simple graphs only. In such a graph the distance d(x, y) between two vertices x, y is defined as the smallest number i such that there is a path consisting of i edges and connecting x and y [8,Section 15.1]. A path between x and y is said to be a geodesic if it is formed by precisely d(x, y) edges. The graph diameter is the greatest distance between two vertices.
A clique is a subset in the vertex set of a graph, where any two distinct elements are adjacent vertices. Using Zorn lemma, we show that every clique is contained in a certain maximal clique.
An embedding of a graph in a graph is an injection of the vertex set of to the vertex set of transferring adjacent and non-adjacent vertices of to adjacent and non-adjacent vertices of , respectively. Surjective embeddings are isomorphisms. Every embedding f sends maximal cliques of to cliques of which are not necessarily maximal, i.e., subsets of maximal cliques. For any distinct maximal cliques X and Y of there exist non-adjacent vertices x ∈ X and y ∈ Y. Then f (x) and f (y) are non-adjacent vertices of and there is no clique containing both f (X ) and f (Y). So, we come to the following observation: every embedding transfers distinct maximal cliques to subsets of distinct maximal cliques.
An embedding is isometric if it preserves the distance between vertices.

Polar spaces
A partial linear space is a pair = (P, L), where P is a non-empty set whose elements are called points and L is a family of proper subsets of P called lines. Every line contains at least two points, and every point belongs to a certain line. Also, for any two distinct points there is at most one line containing them. The points are said to be collinear if such a line exists. A subspace of is a subset S ⊂ P such that for any two collinear points of S the line joining them is contained in S. A subspace is called singular if any two distinct points of this subspace are collinear. The empty set, one-point sets and lines are singular subspaces. Using Zorn lemma, we establish that every singular subspace is contained in a maximal singular subspace. Two subspaces are called incident if one of them is contained in the other.
By [2,13,19,21], a polar space is a partial linear space satisfying the following axioms: (P1) Every line contains at least three points, (P2) There is no point collinear to all points, (P3) For every point and every line the point is collinear to one or all points of the line, (P4) Any chain of mutually distinct incident singular subspaces is finite.
If a polar space has a singular subspace containing more than one line, then all maximal singular subspaces are projective spaces of the same dimension n ≥ 2 and the number n + 1 is called the rank of this polar space. Polar spaces of rank 2 (all maximal singular subspaces are lines) are known as generalized quadrangles. In the case when the rank of a polar space is greater than 2 every singular subspace is a subspace of a certain projective space and its dimension is defined.
Two polar spaces = (P, L) and = (P , L ) are isomorphic if there is a collineation of to , i.e., a bijection α : P → P such that α(L) = L .
All polar spaces of rank ≥ 3 are known [20]. For example, there are polar spaces related to non-degenerate reflexive forms (alternating, symmetric and Hermitian). If such a form is trace-valued and has isotropic subspaces of dimension at least 2, then it defines a polar space: the point set is formed by all one-dimensional isotropic subspaces, the lines are defined by two-dimensional isotropic subspaces, and other isotropic subspaces correspond to singular subspaces of dimension greater than 1.
Consider the (2n)-element set J := {±1, . . . , ±n} and the partial linear space n whose point set is J and whose lines are two-element subsets {i, j} such that j = −i. Then S ⊂ J is a singular subspace of n if and only if for every i ∈ S we have −i / ∈ S. A singular subspace is maximal if it consists of n points. The dimension of a singular subspace S is equal to |S| − 1, and maximal singular subspaces of n are (n − 1)-dimensional. The partial linear space n satisfies the axioms (P2)-(P4), and we say that every partial linear space isomorphic to n is a thin polar space of rank n.
Let = (P, L) be a polar space of rank n. For every subset X ⊂ P the subspace of spanned by X , i.e., the minimal subspace containing X , is denoted by X . If any two distinct points of X are collinear, then this subspace is singular. If a point is collinear to every point of X , then this point is collinear to all points of the subspace X . A subset of P consisting of 2n distinct points p 1 , . . . , p 2n is a frame of if for every i there is unique σ (i) such that p i and p σ (i) are non-collinear. Any k distinct mutually collinear points in a frame span a (k − 1)-dimensional singular subspace. We will use the following remarkable property of frames: for any two singular subspaces there is a frame such that these subspaces are spanned by subsets of the frame. Note that a thin polar space contains the unique frame which coincides with the set of points.
Every rank n polar space satisfies one of the following conditions: (C n ) Every (n − 2)-dimensional singular subspace is contained in at least three maximal singular subspaces, (D n ) Every (n − 2)-dimensional singular subspace is contained in precisely two maximal singular subspaces.
We say that a polar space is of type C n or D n if the corresponding possibility is realized. For example, if a rank n polar space is defined by an alternating or Hermitian form, then it is of type C n . A thin polar space of rank n is of type D n . Other polar spaces of this type will be considered in Sect. 2.5.

Polar Grassmannians
Let = (P, L) be a polar space or a thin polar space of rank n. For every k ∈ {0, 1, . . . , n − 1} we denote by G k ( ) the polar Grassmannian consisting of kdimensional singular subspaces of . Note that G 0 ( ) coincides with P and G n−1 ( ) is formed by maximal singular subspaces.
The polar Grassmann graph k ( ) is the graph whose vertex set is G k ( ). In the case when k ≤ n − 2 two distinct elements of G k ( ) are adjacent vertices of k ( ) if there is a (k + 1)-dimensional singular subspace containing them. Two distinct maximal singular subspaces are adjacent vertices of n−1 ( ) if their intersection is (n − 2)-dimensional. The graph n−1 ( ) is known as the dual polar graph associated with . If is a thin polar space, then we write k (n) instead of k ( ). Note that n−1 (n) is isomorphic to the n-dimensional hypercube graph H n .
For every frame of the set consisting of all k-dimensional singular subspaces spanned by subsets of the frame is called the apartment of G k ( ) associated with this frame. The restriction of the graph k ( ) to every apartment of G k ( ) is isomorphic to k (n). By the frame property given in the previous subsection, for any two elements of G k ( ) there is an apartment containing them. If is a thin polar space, then there is the unique apartment of G k ( ) which coincides with the polar Grassmannian.
For every singular subspace S we denote by [S k the set of all k-dimensional singular subspaces containing S. This set is non-empty only in the case when the dimension of S is not greater than k. Every subset of type is called a line of G n−1 ( ). Each maximal clique in the dual polar graph n−1 ( ) is a line. Now we suppose that k ≤ n−2. Let S and U be a pair of incident singular subspaces such that dim S ≤ k ≤ dim U . Denote by [S, U ] k the set of all X ∈ G k ( ) satisfying S ⊂ X ⊂ U . In the case when S = ∅ we write U ] k instead of [S, U ] k . If dim S = k − 1 and dim U = k + 1, In the case when k = 0 we get a line of .
If 1 ≤ k ≤ n − 3, then there are precisely the following two types of maximal cliques of k ( ): Every star of G n−2 ( ) is a line contained in a certain top, and all maximal cliques of n−2 ( ) are tops. Tops and stars of G 0 ( ) = P are lines and maximal singular subspaces of , respectively.
In the case when 1 ≤ k ≤ n − 2 every subset of type can be naturally identified with the set [S k+i and the polar Grassmann graph i ( S ) coincides with the restriction of the graph k+i ( ) to [S k+i . If A is an apartment of G k ( ) such that S is spanned by a subset of the frame associated with A, then A∩[S k is a frame of S . Conversely, every frame of S can be obtained in this way. Similarly, every apartment of G i ( S ) is the intersection of [S k+i and an apartment of G k+i ( ) such that S is spanned by a subset of the associated frame.

Distance in polar Grassmann graphs
The Grassmann graph k ( ) is connected for every k.
In particular, the diameter of n−1 ( ) is equal to n (the dimension of the empty set is Suppose that there is a point p ∈ X \ Y collinear to all points of Y . Then there exists a point q ∈ Y \ X collinear to all points of X . This follows, for example, from the existence of a frame of containing the point p and such that X and Y are spanned by subsets of this frame. If X ∩ Y is (k − 1)-dimensional, then X and Y are adjacent vertices of k ( ). In the case when we take any k-dimensional singular subspace X 1 spanned by the point q and a (k − 1)dimensional subspace of X containing X ∩ Y and p. Then X and X 1 are adjacent vertices of k ( ) and Step by step, we construct a path Now we suppose that every point of X \ Y is non-collinear to a certain point of Y . Then every point of Y \ X is non-collinear to a certain point of X . We take any frame whose subsets span X and Y and construct a k-dimensional singular subspace X satisfying the following conditions: (1) X and X are adjacent vertices of k ( ), We have So, if k ≤ n − 2, then the diameter of k ( ) is equal to k + 2 and we have the following description of the distance.
The distance between X and Y in k ( ) is equal to m if and only if one of the following possibilities is realized:

collinear to all points of Y and
there is a point of Y \ X collinear to all points of X ;

is non-collinear to a certain point of Y and every point of Y \ X is non-collinear to a certain point of X.
If m = k + 2, then only the second possibility is realized.

Polar spaces of type D n and half-spin Grassmannians
It was noted above that a thin polar space of rank n is of type D n . Let V be a (2n)dimensional vector space over a field. If the characteristic of this field is not equal to 2 and there is a non-degenerate symmetric bilinear form on V whose maximal isotropic subspaces are n-dimensional, then the associated polar space is of type D n .
In the case when the characteristic of the field is equal to 2 we consider a non-defect quadratic form on V such that maximal singular subspaces are n-dimensional. The associated polar space (the points are one-dimensional singular subspaces, and the lines are defined by two-dimensional singular subspaces) is also of type D n . It follows from Tits's description of polar spaces [20] that every polar space of type D n , n ≥ 4 is isomorphic to one of the polar spaces mentioned above. Let = (P, L) be a polar space of type D n (possibly thin) and n ≥ 4. Then G n−1 ( ) can be uniquely decomposed in the sum of two disjoint subsets G + ( ) and G − ( ) such that the distance between any two elements of G δ ( ), δ ∈ {+, −} in the dual polar graph n−1 ( ) is even and the same distance between any S ∈ G + ( ) and U ∈ G − ( ) is odd. These subsets are known as the half-spin Grassmannians.
If is defined by a non-degenerate symmetric bilinear form , then the maximal singular subspaces of are identified with the maximal isotropic subspaces of and the half-spin Grassmannians are the orbits of the action of the group O + ( ) (the orthogonal group formed by elements with determinant 1) on the set of all maximal isotropic subspaces. Every element of O( ) \ O + ( ) induces a collineation of which maps one of the half-spin Grassmannians to the other. The same holds for the case when is defined by a quadratic form. So, collineations of sending G + ( ) to G − ( ) always exist.
Suppose that n = 4 and δ ∈ {+, −}. For every line L ∈ L, the set [L δ consisting of all elements of G δ ( ) containing L is called a line of G δ ( ). The half-spin Grassmannian G δ ( ) together with the family of all such lines is a polar space of type D 4 . We denote this polar space by δ . The polar spaces + and − are isomorphic (every collineation of transferring G + ( ) to G − ( ) induces a collineation between these polar spaces). The half-spin Grassmannians corresponding to δ are the point set P and G −δ ( ), where −δ is the complement of δ in the set {+, −}. The associated polar spaces are and −δ . Therefore, is isomorphic to both + and − . Since there is a natural one-to-one correspondence between lines of the polar spaces and δ , every collineation from to δ induces a bijective transformation of G 1 ( ). This transformation is an automorphism of the graph 1 ( ).
Let α be a collineation from to −δ . It induces collineations of the polar spaces + and − to the polar spaces associated with the half-spin Grassmannians of −δ . So, we get a collineation of δ to or δ . Since there are collineations of −δ transferring to δ , we can suppose that α induces a collineation of δ to itself.
The automorphism g of 1 ( ) induced by α has the following properties: See [13,Section 4.6] for the details.

Main results
From this moment we suppose that = (P, L) is a polar space or a thin polar space of rank n and = (P , L ) is a polar space of rank n . Let f : P → P be a collinearity-preserving injection, i.e., f sends collinear and non-collinear points of to collinear and non-collinear points of , respectively. We now show that f transfers every frame of to a subset in a frame of .
If F is a frame of , then for every point p ∈ f (F) there is a unique point of f (F) non-collinear to p. This means that n ≤ n and f (F) is a frame of if n = n . In the case when n > n we consider the set formed by all points of collinear with all points of f (F). If n − n ≥ 2, then this is a polar space of rank n − n and f (F) together with any frame of this polar space gives a frame of . If n − n = 1, then our set consists of mutually non-collinear points and f (F) together with any pair of such points defines a frame of .
Since every singular subspace S of is spanned by a subset of a certain frame of , the dimension of the singular subspace f (S) is equal to the dimension of S. It is clear that f is an isometric embedding of 0 ( ) in 0 ( ) and for every k ∈ {1, . . . , n − 1} the mapping is an isometric embedding of k ( ) in k ( ). If n = n and is a thin polar space, then the image of this mapping is an apartment of G k ( ). Now we suppose that n ≤ n and take any m-dimensional singular subspace S of such that m ≤ n − n − 1 (this subspace is empty if n = n ). Then S is a polar space of rank n − m − 1 ≥ n (in the case when S is empty, this polar space coincides with ). Every collinearitypreserving injection of to S induces an isometric embedding of k ( ) in k ( S ).
For every k ∈ {0, 1, . . . , n − 1} this mapping can be considered as an isometric Remark 1 Suppose that the polar spaces and are embedded in the projective spaces associated with vector spaces V and V , respectively. Every collineation from to is induced by a semilinear isomorphism of V to V [9, Chapter III]. Similarly, every collinearity-preserving mapping from to can be obtained from a certain semilinear mapping l : V → V (the proof is a modification of the proof given in [9,Chapter III]). Note that the homomorphism between division rings associated with l is not necessarily surjective. In the general case, we cannot state that l is a strong semilinear embedding (a semilinear injection transferring any collection of linearly independent vectors to linearly independent vectors). However, this statement is obvious if our polar spaces are of type D n or for the symplectic polar spaces.
In this paper we will investigate isometric embeddings of the polar Grassmann graph k ( ) (this graph coincides with k (n) if is a thin polar space) in the polar Grassmann graph k ( ). We start from the following simple observation that will be proved in Sect. 4.

Remark 2
The diameter of 0 ( ) is equal to 2, and every embedding of this graph is isometric.
All isometric embeddings of the dual polar graph n−1 ( ) in the dual polar graph n −1 ( ) are described in [14]. The existence of such embeddings implies that the diameter of n−1 ( ) is not greater than the diameter of n −1 ( ), i.e., n ≤ n . By [14,Theorem 2], the image of every isometric embedding of the n-dimensional hypercube graph H n = n−1 (n) in the dual polar graph n −1 ( ) is an apartment of G n−1 ( S ), where S is an (n − n − 1)-dimensional singular subspace of . Using this result and [13,Theorem 4.17] the author shows that every isometric embedding of n−1 ( ) in n −1 ( ) is induced by a collinearity-preserving injection from to S , where, as above, S is an (n − n − 1)-dimensional singular subspace of [14,Theorem 3]. We will consider the case when our polar Grassmann graphs both are formed by non-maximal singular subspaces. The first result concerns the case when n ≥ 5 and 1 ≤ k ≤ n − 4. It must be pointed out that in Theorem 1 there is no assumption concerning n and k . The case when n ≥ 4 and k = n − 3 is different.

Theorem 2 Suppose that n ≥ 4 and f is an isometric embedding of n−3 ( ) in k ( ).
If is a polar space of type C n , then and there is a (k − n + 2)-dimensional singular subspace S of such that the image of f is contained in [S k and f is induced by a collinearity-preserving injection of to S .
In the case when is a polar space of type D n the following assertions are fulfilled: (1) If n = 4, then 1 ≤ k ≤ n − 3 and there is a (k − 2)-dimensional singular subspace S of such that the image of f is contained in [S k . Also, there is an automorphism g of 1 ( ) (possibly the identity) such that the composition f g is induced by a collinearity-preserving injection from to S .
(2) If n ≥ 5 and k = n − 3, then n ≤ n and there is a (n − n − 1)-dimensional singular subspace S of such that the image of f is contained in [S k and f is induced by a collinearity-preserving injection from to S . Theorem 2 does not contain any assumption concerning n and k except the case when is a polar space of type D n , n ≥ 5. In this special case we can describe isometric embeddings of n−3 ( ) in n −3 ( ) only (see Remark 3 for the reasons).
Our third result covers the case when n = n and k = k = n − 2.
Theorem 3 If n = n , then every isometric embedding of n−2 ( ) in n−2 ( ) is induced by a collinearity-preserving injection of to .
We cannot prove the same statement for isometric embeddings of n−2 ( ) in n −2 ( ) if n < n (see Remark 4 for the reasons). As a direct consequence of the above results, we get the following characterization of apartments in polar Grassmannians.

Corollary 1 The image of every isometric embedding of k (n) in k ( ) is an apartment of G k ( ).
Proof For k = 0 the statement follows directly from the frame definition. The case k = n − 1 was considered in [14, Theorem 2]. If 1 ≤ k ≤ n − 2, then we apply Theorems 1-3 to isometric embeddings of k (n) in k ( ).
Also, Theorems 1 and 2 imply that for the residue k ( S ), where S is a singular subspace, the same statement holds with some restrictions on k and the type of . k ≤ n − 2. If 1 ≤ k ≤ n − 3, then there are the following two types of triangles: star-triangles contained in stars and top-triangles contained in tops [13,Lemma 4.10]. If S 1 , S 2 , S 3 ∈ G k ( ) form a star-triangle, then dim(S 1 ∩ S 2 ∩ S 3 ) = k − 1 and dim S 1 , S 2 , S 3 = k + 2.
In the case when S 1 , S 2 , S 3 ∈ G k ( ) form a top-triangle we have Each triangle of is a star-triangle, and G n−2 ( ) contains only top-triangles. Proof In this case, every maximal clique of k ( ) contains a triangle. On the other hand, the intersection of two maximal cliques of different types is empty or a line. Lemma 2 gives the claim.

Proof of Proposition 1
Suppose that f is an embedding of 0 ( ) in m ( ) (see Remark 2) and n ≥ 4.
Let M be a maximal singular subspace of . We take any maximal singular subspace M such that M ∩ M is (n −2)-dimensional.
So, the image of f is contained in [S m and f is a collinearity-preserving injection of to S . The rank of S is equal to n − m, and it is not less than n which implies that m ≤ n − n.

Technical result
Let f be an isometric embedding of k ( ) in k ( ) and 1 ≤ k ≤ n − 3. Then maximal cliques of k ( ) are stars and tops and there exist pairs of distinct maximal cliques whose intersections contain more than one element. In the case when k ≥ n −2 there is only one type of maximal cliques in k ( ) (tops if k = n − 2 and lines if k = n − 1) and the intersection of any two distinct maximal cliques contains at most one element. It was noted in Sect. 2.1 that f sends distinct maximal cliques to subsets of distinct maximal cliques. This guarantees that k ≤ n − 3. Also, the existence of isometric embeddings of k ( ) in k ( ) implies that the diameter of k ( ) is not greater than the diameter of k ( ). By Sect. 2.4. the diameters of these graphs are equal to k + 2 and k + 2, respectively. Therefore, k ≤ k .

Proposition 2 If f transfers stars to subsets of stars, then
n − k ≤ n − k and there exists a (k − k − 1)-dimensional singular subspace S of such that the image of f is contained in [S k and f is induced by a collinearity-preserving injection of to S . Proposition 2 will be proved in several steps. Our first step is the following. The intersection of two distinct big stars contains at most one element. This implies that for any

Lemma 5 The mapping f k−1 is injective.
Proof Let S and U be distinct elements of G k−1 ( ). We take any frame of such that S and U are spanned by subsets of this frame. The associated apartment A ⊂ G k ( ) contains X ∈ [S k and Y ∈ [U k satisfying d(X, Y ) ≥ 3. Indeed, if the dimension of S ∩ U is less than k − 2, then we choose any X ∈ A ∩ [S k and Y ∈ A ∩ [U k such that In the case when S ∩ U is (k − 2)-dimensional we require in addition that every point of X \ Y is non-collinear to a certain point of Y . See Lemma 1. If which contradicts the fact that f is an isometric embedding of k ( ) in k ( ).
For every U ∈ G k ( ) we have i.e., f k−1 transfers tops to subsets of tops, which implies that f k−1 sends adjacent vertices of k−1 ( ) to adjacent vertices of k −1 ( ). However, we cannot state that non-adjacent vertices of k−1 ( ) go to non-adjacent vertices of k −1 ( ).
Suppose that k ≥ 2. Then f k−1 transfers maximal cliques of k−1 ( ) (stars and tops) to subsets of maximal cliques of k −1 ( ). We do not show that f k−1 is an isometric embedding of k−1 ( ) in k −1 ( ). It is sufficient to prove the following.

Lemma 6 f k−1 transfers stars to subsets of stars.
Proof Suppose that there exists a star S ⊂ G k−1 ( ) whose image is not contained in a star of G k −1 ( ). Then f k−1 (S) is a subset in a certain top T ⊂ G k −1 ( ). We choose distinct U 1 , U 2 ∈ G k ( ) such that for every i = 1, 2 the top U i ] k−1 intersects the star S in a line. Then the intersection of f (U i )] k −1 and T contains more than one element. This is possible only in the case when the tops f (U i )] k −1 and T are coincident. Hence f (U 1 ) = f (U 2 ) which contradicts the fact that f is injective.

Lemma 7 There is a sequence of injections
Proof Using Lemmas 5, 6 and the arguments from the proof of Lemma 4 we show that f k−1 transfers big stars to subsets of big stars and the image of every big star of G k−1 ( ) is contained in a unique big star of G k −1 ( ). So, f k−1 induces a mapping If S and U are distinct elements of G k−2 ( ), then there exist X ∈ [S k and Y ∈ [U k satisfying d(X, Y ) ≥ 4 (as in the proof of Lemma 5 we take a frame of such that S and U are spanned by subsets of this frame and choose X, Y in the associated apartment of G k ( )). It is clear that If f k−2 (S) coincides with f k−2 (U ), then the dimension of the intersection of f (X ) and f (Y ) is not less than k − 2. Lemma 1 shows that and f is not an isometric embedding of k ( ) in k ( ). Therefore, f k−2 is injective.
The mapping f k−2 transfers tops to subsets of tops. As in the proof of Lemma 6 we show that f k−2 sends stars to subsets of stars if k ≥ 3.
Step by step, we construct the required sequence.

The inclusion from Lemma 7 implies that
then U ] 0 is the union of all S] 0 such that S ∈ U ] i−1 and the latter inclusion can be proved by induction.
As in the proof of Proposition 1 (Sect. 4.2) we show that f 0 ( p) belongs to [S k −k for every point p ∈ P. If U ∈ G k ( ) and p is a point of U , then Therefore f is an isometric embedding of k ( ) in k ( S ) and f 0 is an injection of the point set P to [S k −k transferring lines of to subsets in lines of S .

Lemma 9
Let U ∈ G i ( ) and i ≤ k. If U is spanned by points p 1 , . . . , p i+1 , then Proof We prove the statement by induction. The case when i = 0 is trivial. Suppose that i ≥ 1, and consider the (i − 1)-dimensional singular subspaces M and N spanned by p 1 , . . . , p i and p 2 , . . . , p i+1 , respectively. By the inductive hypothesis f i−1 (M) and f i−1 (N ) are spanned by f 0 ( p 1 ), . . . , f 0 ( p i ) and f 0 ( p 2 ), . . . , f 0 ( p i+1 ), respectively. The required statement follows from the fact that f i−1 (M) and f i−1 (N ) are distinct (k −k +i −1)-dimensional singular subspaces contained in the (k −k +i)dimensional singular subspace f i (U ).
Our last step is the following.

Lemma 10 f 0 is a collinearity-preserving injection of to S .
Proof Since f 0 transfers lines of to subsets in lines of S , we need to show that f 0 sends non-collinear points of to non-collinear points of S .
Let p and q be non-collinear points of . Consider a frame where every p i is non-collinear to q i . Denote by X and Y the k-dimensional singular subspaces spanned by p 1 , . . . , p k+1 and q 1 , . . . , q k+1 , respectively. Then Lemma 1). By Lemma 9 f (X ) and f (Y ) are k-dimensional subspaces of S spanned by respectively. The point p 1 is collinear to Lemma 1). The latter is impossible, since f is an isometric embedding of k ( ) in k ( ). Thus f 0 ( p) and f 0 (q) are non-collinear points of S .
So, f 0 is a collinearity-preserving injection of to S . It follows from Lemma 9 that f (U ) coincides with f 0 (U ) for every U ∈ G k ( ), i.e., f is induced by f 0 .
The rank of S is equal to n − k + k. The existence of collinearity-preserving injections of to S implies that n − k + k ≥ n. This completes the proof of Proposition 2.

Proof of Theorem 1
Let f be as in the previous subsection. We need to show that f transfers stars to subsets of stars if n ≥ 5 and k ≤ n − 4.
Suppose that there is a star S ⊂ G k ( ) such that f (S) is contained in a top of G k ( ). If X, Y, Z ∈ S form a triangle, then their images form a top-triangle. The corresponding top is the unique maximal clique of k ( ) containing f (X ), f (Y ), f (Z ). On the other hand, the singular subspace spanned by X, Y, Z is (k + 2)-dimensional. Since k ≤ n − 4, this singular subspace is not maximal. This guarantees the existence of a star S containing X, Y, Z and different from S. By the observation from Sect. 2.1 f (S) and f (S ) are subsets of distinct maximal cliques of k ( ). Each of these cliques contains f (X ), f (Y ), f (Z ), and we get a contradiction.

Proof of Theorem 2
In this section we suppose that f is an isometric embedding of n−3 ( ) in k ( ) and n ≥ 4. By Sect. 4.3 we have n − 3 ≤ k ≤ n − 3.

Regular pairs of triangles
be triangles in G k ( ) such that ∩ = ∅. We say that these triangles form a regular pair if S i and S j are adjacent vertices of k ( ) only in the case when i = j; in other words, every vertex from each of these triangles is adjacent to precisely two vertices of the other triangle. In this case, one of the following possibilities is realized: (1) There are (k + 2)-dimensional singular subspaces U and U whose intersection S is (k − 1)-dimensional and , are star-triangles contained in [S, U ] k and [S, U ] k , respectively. Note that for every point q ∈ U \ S there is a point of U non-collinear to q. Similarly, for every point q ∈ U \ S there is a point of U non-collinear to q .
(2) One of the triangles is a star-triangle, and the other is a top-triangle. For example, if is a star-triangle and is a top-triangle, then the singular subspace S 1 , S 2 , S 3 is (k + 1)-dimensional and there is a point p / ∈ S 1 , S 2 , S 3 collinear to all points of S 1 , S 2 , S 3 and such that Note that all elements of our triangles are contained in the (k + 2)-dimensional singular subspace spanned by p and S 1 , S 2 , S 3 .
Indeed, two top-triangles do not form a regular pair. So, at least one of them is a star-triangle. If our triangles both are star-triangles, then is a (k − 1)-dimensional subspace, we denote it by S and get (1). If one of the triangles is a star-triangle and the other is a top-triangle, then a direct verification shows that the case (2) is realized.
It is clear that f transfers regular pairs of triangles to regular pairs of triangles.

Proof of Theorem 2 for n = 4
Suppose that n = 4. Then f is an isometric embedding of 1 ( ) in k ( ). A maximal singular subspace U ∈ G 3 ( ) is said to be special if there exists a point This statement is proved for the case when k = 1 and n = 4 [13,Lemma 4.15]. Now we show that the same arguments work in the general case.
We take any two-dimensional singular subspace S ⊂ U which does not contain the point p. Consider a regular pair of triangles Since f ( ) is a top-triangle, the triangles f ( ) and f ( ) form a regular pair of type (2). Therefore f ( ) is a star-triangle and f ( S] 1 ) is contained in a certain star Note that S ⊂ U and T ⊂ S . Let q be a point belonging to U \ {p}. We choose a two-dimensional singular subspace S ⊂ U which does not contain p and q. It was established above that f ( S] 1 ) is a subset in a certain star (1). Consider a regular pair of triangles ⊂ [q, U ] 1 , and ⊂ S] 1 .
Then q / ∈ S and f ( ) is a star-triangle. Suppose that the triangles f ( ) and f ( ) form a regular pair of type (1). Let S be a two-dimensional singular subspace of U containing the point p. We take any point q ∈ U \ S. Then f ([q, U ] 1 ) is contained in a top. As above, we establish that f ( S] 1 ) is a subset in a star.

Lemma 12
Let U, Q ∈ G 3 ( ). If U is special and dim(U ∩ Q) = 2, then Q is not special.
Proof We take two distinct lines L 1 , L 2 ⊂ U ∩ Q and consider star-triangles 1 and Q] 1 , respectively. The lines L 3 and L 3 have a common point; on the other hand, they contain non-collinear points which means that the distance between them in 1 ( ) is equal to 2. Thus form a triangle. This is not a top-triangle (otherwise, f (L 1 ) belongs to the top contain- form a star-triangle and f (L 2 ) contains the (k − 1)-dimensional singular subspace .
So, f (L 1 ) and f (L 2 ) intersect f (L) in two distinct (k − 1)-dimensional singular subspaces which means that f ( ) is a top-triangle. Then f sends the star containing to a subset in a top, i.e., Q is special.
Let X and Y be adjacent vertices of 3 ( ). Suppose that is a polar space of type C 4 . Then the line joining X and Y contains a certain Z ∈ G 3 ( ) distinct from X, Y . We take any T ∈ G 3 ( ) which does not belong to this line and such that Z , T are adjacent vertices of 3 ( ). Then If X is special, then T is special by Lemma 13. We apply Lemma 13 to T, Y and establish that Y is special. The latter is impossible by Lemma 12. Therefore the existence of special maximal singular subspaces implies that is a polar space of type D 4 . Also, it follows from Lemmas 12 and 13 that all special maximal singular subspaces form one of the half-spin Grassmannians G δ ( ), δ ∈ {+, −}. Every two-dimensional singular subspace of is contained in a certain element of G δ ( ). Thus f transfers every top to a subset in a star. Let g be the automorphism of 1 ( ) described in Sect. 2.5. Then the composition f g is an isometric embedding of 1 ( ) in k ( ) transferring stars to subsets of stars. We apply Proposition 2 again.

Proof of Theorem 2 for n ≥ 5
Let n ≥ 5. Suppose that there is a star is contained in a top. The singular subspace S is (n − 4)-dimensional, and we take any (n −5)-dimensional singular subspace T ⊂ S. The rank of the polar space T is equal to 4, and G 1 ( T ) coincides with [T n−3 . The star [S, U ] n−3 is contained in [T n−3 , i.e., it is a star of G 1 ( T ). The restriction of f to [T n−3 is an isometric embedding of 1 ( T ) in k ( ). By Sect. 5.2, T is a polar space of type D 4 which implies that is a polar space of type D n . Thus f transfers stars to subsets of stars if is a polar space of type C n . As above, we use Proposition 2. Now we consider the case when k = n − 3 and show that f sends every star to a subset in a star.
Suppose that S ⊂ G n−3 ( ) is a star such that f (S) is contained in a top. We take any top U ] n−3 , U ∈ G n−2 ( ) intersecting S in a line. Since the intersection of two distinct tops contains at most one element, f ( U ] n−3 ) cannot be in a top. Hence it is contained in a certain star If X 1 , X 2 , X 3 ∈ U ] n−3 form a triangle, then their images form a star-triangle and The dimension of S := X 1 ∩ X 2 ∩ X 3 is equal to n − 5. We choose Y ∈ G n−3 ( ) satisfying the following conditions: • Y ∩ U = S, • There is a point of Y \U collinear to all points of U (note that the singular subspace U is not maximal).
By Lemma 1, d(X i , Y ) = 2 for every i and we have d(Z , Y ) = 3 for every Z ∈ U ] n−3 which does not contain S. We want to show that This contradicts the fact that f is an isometric embedding of n−3 ( ) in n −3 ( ), and we get the claim. This point belongs to U . Hence p ∈ f (Y ) \ S is collinear to all points of every element of [S , U ] n −3 . Since f (Y ) ∩ S is (n − 5)-dimensional, we get (2) again (see Lemma 1).
Now we suppose that f (X i ) ∩ f (Y ) is (n − 5)-dimensional for every i. If this subspace is contained in S for a certain i, then all f (X i ) ∩ f (Y ) are coincident with f (Y ) ∩ S and the latter subspace is (n − 5)-dimensional. Since the distance between f (X i ) and f (Y ) is equal to 2, for every i there exists a point collinear to all points of f (Y ) (see Lemma 1). If one of the points p i belongs to S , then it is contained in every element of [S , U ] n −3 . This implies (2), since f (Y ) ∩ S is (n − 5)-dimensional (see Lemma 1). Suppose that p i ∈ f (X i ) \ S for every i, and consider the two-dimensional singular subspace T spanned by p 1 , p 2 , p 3 (the dimension of T is equal to 2, since the (n − 1)-dimensional singular subspace U is spanned by T and the (n − 4)-dimensional singular subspace S ). Every point of T is collinear to all points of f (Y ) and T ∩ S = ∅. Each X ∈ [S , U ] n −3 has a non-empty intersection with T , i.e., there is a point of X \ S collinear to all points of f (Y ). As above, we get (2). Suppose that the distance between g(X ) and g(Y ) is less than n. Then g(X ) and g(Y ) have a non-empty intersection. The distance between f (A) and f (D) is equal to n − 1 and, by Lemma 1, we have two possibilities. In each of these cases, there is a point q ∈ f (D) collinear to all points of f (A). Let p ∈ g(X ) ∩ g(Y ). By Lemma 14 g(X ) is spanned by f (A) and p. Since q is collinear to p and all points of f (A), it is collinear to all points of g(X ). The latter means that q ∈ g(X ) (recall that g(X ) is a maximal singular subspace of ). Then q belongs to g(X ) ∩ g(Y ), which contradicts Lemma 14.
Thus g(X ) ∩ g(Y ) = ∅, i.e., the distance between g(X ) and g(Y ) is equal to n. So, g is an isometric embedding of n−1 ( ) in n−1 ( ). By [14] it is induced by a collinearity-preserving injection from to . If S ∈ G n−2 ( ) is the intersection of X, Y ∈ G n−1 ( ), then f (S) coincides with g(X ) ∩ g(Y ). This implies that f is induced by the same collinearity-preserving injection of to .