Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 261-266.

m-Point Invariants of Real Geometries

Roland Höfer

Mathematisches Seminar der Universität Hamburg Bundesstraße 55, D-20146 Hamburg, Germany

Abstract. Let V be a finite dimensional vector space over a field K with a nondegenerate symmetric, alternating, or hermitian scalar product. We characterize the orbits of m-tuples of vectors of V under the group of isometries of V by the Gram matrix and an additional subspace of K^{m}. This characterization will then be applied to determine all m-point invariants of real euclidean, spherical, hyperbolic, and de-Sitter geometries. MSC 1991: 15A63, 51M10, 83A05

1. Introduction

Let P be a set and Γ a group of permutations of P. The pair (P, Γ) is called a geometry, the elements of P are called points, the elements of Γ are called motions. Let m be a natural number. An m-point invariant ι of (P, Γ) maps m-tuples of points to an arbitrary non-empty set S and satisfies the functional equation

ι(x_{1}, …, x_{m}) = ι(γ(x_{1}), …, γ(x_{m}))

for all x_{1}, …, x_{m} ∈ P, γ ∈ Γ. Let S_{0} be the set of orbits of m-tuples of points under Γ. Let ω : P^{m} → S_{0} map any m-tuple of points to its orbit. Let ψ : S_{0} → S be an arbitrary mapping, then ι := ψ ∘ ω : P^{m} → S is an m-point invariant; on the other hand, for any m-point invariant ι : P^{m} → S there exists a uniquely determined mapping ψ : S_{0} → S such that ι = ψ ∘ ω. We are looking for a concrete characterization of the orbits.

W. Blaschke has determined all invariants of m-tuples of vectors in a real finite dimensional orthogonal space in the following way [2, §10]:

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

262 Roland Höfer: m-Point Invariants of Real Geometries

“Man bestimme zuerst die höchstmögliche Zahl r von linear unabhängigen Vektoren unter den gegebenen. Dann greife man irgendein System von r linear unabhängigen ‘Grundvektoren’ aus ihnen heraus und kombiniere alle übrigen aus diesen linear. Das vollständige System der unabhängigen Invarianten ist dann gegeben durch die Skalarprodukte der Grundvektoren und die Koeffizienten der gebildeten Linearkombinationen.”

A similar result for more general spaces is obtained in Theorem 1. The orbit of m vectors is characterized by their Gram matrix and an additional subspace. Of course, if the vectors are linearly dependent, then the components of their Gram matrix are dependent; in Theorem 2 we determine all possible Gram matrices and related subspaces in an important case. The 2- and 3-point invariants of some real geometries have been examined in [1]. Theorem 1 and 2 are applied to give all m-point invariants of real euclidean, spherical, hyperbolic, and de-Sitter geometries.

2. Transitivity properties

In this section let V be a finite dimensional left- vector space over a (skew-) field K and let α be an automorphism of K. Let (·, ·) : V × V → K be an α-scalar product (i.e. for all a_{1}, a_{2}, a_{3} ∈ K, v_{1}, v_{2}, v_{3} ∈ V,

(a_{1} v_{1} + a_{2} v_{2}, v_{3}) = a_{1} (v_{1}, v_{3}) + a_{2} (v_{2}, v_{3}),
(v_{1}, a_{2} v_{2} + a_{3} v_{3}) = α(a_{2}) (v_{1}, v_{2}) + α(a_{3}) (v_{1}, v_{3}) ),

which is non-degenerate and which is symmetric, alternating, or hermitian; we assume char K ≠ 2, if (·, ·) is symmetric.

Theorem 1. Let m ∈ ℕ and x_{1}, …, x_{m}, x′_{1}, …, x′_{m} ∈ V. There is an isometry of (V, (·, ·)) which takes x_{i} to x′_{i}, i = 1, …, m, iff the Gram matrices of x_{1}, …, x_{m} and x′_{1}, …, x′_{m} are equal,

(x_{i}, x_{j}) = (x′_{i}, x′_{j}),  (1)

and

∑_{i=1}^{m} λ_{i}x_{i} = 0 ⇔ ∑_{i=1}^{m} λ_{i}x′_{i} = 0  (2)

for all λ = (λ_{1}, …, λ_{m}) ∈ K^{m}.

In order to prove Theorem 1 we need a theorem of Witt (see e.g. [3]): Let U_{1}, U_{2} be subspaces of V and let φ : U_{1} → U_{2} be an isometry, i.e. φ is a linear isomorphism which is metric, (φ(x), φ(y)) = (x, y) for all x, y ∈ U_{1}. Then φ can be extended to an isometry f of (V, (·, ·)), f|_{U_{1}} = φ.

Proof of Theorem 1. If there is an isometry f which takes x_{i} to x′_{i}, i = 1, …, m, then the conditions (1) and (2) are satisfied. Now let x_{1}, …, x_{m} and x′_{1}, …, x′_{m} satisfy (1) and (2). To x_{1}, …, x_{m} corresponds a linear map g : K^{m} → V,

g(λ) = ∑_{i=1}^{m} λ_{i}x_{i}.  (3)

To x′_{1}, …, x′_{m} we define g′ : K^{m} → V analogously. Then

(g(λ), g(μ)) = (g′(λ), g′(μ)) for all λ, μ ∈ U,  (1’)
ker g = ker g′.  (2’)

Let v_{1}, …, v_{r} be a basis of im g. Take u_{1}, …, u_{r} ∈ U with g(u_{i}) = v_{i}, i = 1, …, r. Then u_{1}, …, u_{r} are linearly independent and U = span{u_{1}, …, u_{r}} ⊕ ker g. From (2’) follows that the images v′_{1}, …, v′_{r} of u_{1}, …, u_{r} under g′ are a basis of im g′. The linear map φ : im g → im g′ defined by φ(v_{i}) = v′_{i}, i = 1, …, r, is an isometry because of (1’). From Witt’s theorem there is an isometry f : V → V which satisfies f|_{im g} = φ, thus g′ = φ ∘ g = f ∘ g, and x′_{i} = f(x_{i}), i = 1, …, m.

3. A characterization of the orbits

Let G be an m × m matrix and let W be a subspace of K^{m}. In view of Theorem 1 we may ask: Are there x_{1}, …, x_{m} ∈ V such that

G is the Gram matrix of x_{1}, …, x_{m}  (4)

and such that, for all λ ∈ K^{m},

∑_{i=1}^{m} λ_{i}x_{i} = 0 ⇔ (λ_{1}, …, λ_{m}) ∈ W  (5)

is satisfied? We give necessary and sufficient conditions in an important case:

Theorem 2. Let K be an ordered commutative field whose positive numbers are squares. Let m, n ∈ ℕ and k ∈ {0, …, n}. Let V be an n-dimensional vector space over K with a nondegenerate, symmetric bilinear form (·, ·)_{V} and signature (k, n − k). Let G be a symmetric m × m-matrix with entries in K and with s positive and t negative eigenvalues, s + t ≤ m. Let W be a subspace of K^{m} and let δ := m − s − t − dim W. There are x_{1}, …, x_{m} ∈ V with (4) and (5) iff

s + δ ≤ k, t + δ ≤ n − k, W G := {wG | w ∈ W } = 0.  (6)

Proof. We define a symmetric bilinear form (·, ·)_{K^{m}} on K^{m} by (λ, μ)_{K^{m}} := λGμ^{T} ∈ K. The signature of (K^{m}, (·, ·)_{K^{m}}) is (s, t), and the radical of K^{m} is

rad K^{m} := {λ ∈ K^{m} | (λ, μ)_{K^{m}} = 0 ∀μ ∈ K^{m}}
= {λ ∈ K^{m} | λG = 0}.

We take an orthogonal basis u_{1}, …, u_{m} of K^{m} such that (u_{i}, u_{i})_{K^{m}} = 1 if i ≤ s, (u_{i}, u_{i})_{K^{m}} = −1 if s < i ≤ s + t, and such that u_{s+t+1}, …, u_{m} is a basis of the radical of K^{m}. Let x_{1}, …, x_{m} ∈ V satisfy (4) and (5). The linear map g : K^{m} → V defined by (3) is metric, and W = ker g < rad K^{m}. The space V_{1} spanned by g(u_{1}), …, g(u_{s+t}) is a regular subspace of V with signature (s, t). The space V_{1}^{⊥} orthogonal to V_{1} is a regular subspace of V and has signature (k − s, n − k − t). The space V_{2} spanned by g(u_{s+t+1}), …, g(u_{m}) is an isotropic subspace of V_{1}^{⊥}. Hence δ = dim V_{2} ≤ min{k − s, n − k − t}, and (6) is satisfied. We now assume that (6) is satisfied. Let v_{1}, …, v_{n} ∈ V be an orthogonal basis of V such that (v_{i}, v_{i})_{V} = 1 if i ≤ k and (v_{i}, v_{i})_{V} = −1 if i > k. Take a basis w_{1}, …, w_{m−s−t} of rad K^{m} such that w_{δ+1}, …, w_{m−s−t} is a basis of W. We define the linear map g : K^{m} → V by

\begin{matrix}u_{i} ↦ & v_{i}, & i = 1, …, s, \\ u_{s+i} ↦ & v_{k+i}, & i = 1, …, t, \\ w_{i} ↦ & v_{s+i} + v_{k+t+i}, & i = 1, …, δ, \\ w_{δ+i} ↦ & 0, & i = 1, …, m − s − t − δ. \end{matrix}

g is metric, its kernel is W. Hence the vectors x_{1}, …, x_{m} ∈ V defined by (3) satisfy (4) and (5).

Corollary 1. The pairs (G, W) in Theorem 2 which satisfy (6) correspond to the orbits of m-tuples of vectors of V under the isometries of (V, (·, ·)_{V}).

Corollary 2. Let K be an ordered commutative field whose positive numbers are squares. Let V be an n-dimensional vector space over K with a symmetric bilinear form (·, ·) and signature (k, n − k). Let G be the Gram matrix of x_{1}, …, x_{m} ∈ V and let s and t be the numbers of positive and negative eigenvalues of G, respectively.

a) G alone characterizes the orbit of (x_{1}, …, x_{m}) under the isometries of (V, (·, ·)) (i.e., whenever G is the Gram matrix of x′_{1}, …, x′_{m} ∈ V, then there is an isometry which takes x_{i} to x′_{i}, i = 1, …, m) if, and only if, det G ≠ 0 or s = k or t = n − k.

b) G and the maximal number r of independent vectors in {x_{1}, …, x_{m}} characterize the orbit of (x_{1}, …, x_{m}) if and only if r = m or r = rank G.

Proof. Let κ := {λ ∈ ℝ^{m} | λG = 0}.

a) If m − s − t = 0, then W = {0} is the only subspace of ℝ^{m} with (6). If s = k or t = n − k, then W = κ is the only subspace of ℝ^{m} with (6). If m − s − t > 0 and s < k and t < n − k, then any (m − s − t − δ)-dimensional subspace W of κ satisfies (6), δ ∈ {0, 1}.

b) Let W be defined by (5). Then W is a subspace of κ. There is no other subspace of κ which has the same dimension as W, iff m − r = dim W ∈ {0, m − s − t}.

In the case K = ℝ, Corollary 2b) is stated in [2, §32], the right side of the equivalence is: r = n or r = m or r = rank G. We note that r = n implies r = rank G.

4. Euclidean vector spaces

Let m, n ∈ ℕ. Let (V, (·, ·)) be an n-dimensional real euclidean vector space. Its signature is (n, 0). The Gram matrix of m vectors x_{1}, …, x_{m} ∈ V satisfies

G = G^{T} is positive semi-definite, rank G ≤ n.  (7)

On the other hand, any real m×m matrix which satisfies (7) is the Gram matrix of m vectors of V. Thus all m × m matrices with (7) correspond to the orbits of m-tuples of vectors under the isometries of V; cf. e.g. [3, §II.12].

A motion of n-dimensional euclidean geometry E_{n} = V is a mapping E_{n} → E_{n}, x ↦ g(x) + t where g is an isometry of (V, (·, ·)) and t ∈ V. It is trivial that the orbit of m + 1 vectors x_{0}, …, x_{m} ∈ E_{n} under the group of motions of E_{n} corresponds to the Gram matrix G of x_{1} − x_{0}, …, x_{m} − x_{0}, which satisfies (7).^{1}

Let n ≥ 2. The set of points of (n − 1)-dimensional spherical geometry is 𝕊^{n−1} := {x ∈ V | (x, x) = 1}. The spherical motions are the restrictions of the isometries of (V, (·, ·)) to 𝕊^{n−1}. The spherical distance between x, y ∈ 𝕊^{n−1} is d(x, y) = arccos(x, y). The orbit of m points x_{1}, …, x_{m} ∈ 𝕊^{n−1} corresponds to their Gram matrix G = (g_{ij}) = (cos d(x_{i}, x_{j})), which satisfies (7) and

g_{11} = … = g_{mm} = 1.  (8)

If a real m × m matrix G = (g_{ij}) satisfies (7) and (8), then it is the Gram matrix of m spherical points.

5. Minkowski vector spaces

Let m, n ∈ ℕ, n ≥ 2. Let (V, (·, ·)) be an n-dimensional real vector space whose signature is (n − 1, 1). A vector v ∈ V \ {0} is called timelike, lightlike, spacelike if (v, v) < 0, = 0, > 0, respectively. By 𝔏 we denote the set of all subspaces of V which contain lightlike, but no timelike vectors. According to Theorem 1 and 2, the pairs (G, W) which satisfy (6) correspond to the orbits of m-tuples of vectors under the isometries of V; we now have δ ∈ {0, 1}. Let κ := {λ ∈ ℝ^{m} | λG = 0}. Let x_{1}, …, x_{m} be vectors with (4) and (5), and let X < V be the subspace spanned by x_{1}, …, x_{m}. In the case δ = 0 we have X ∉ 𝔏, and W = κ is determined by G. In the case δ = 1 we have X ∈ 𝔏, and W is a hyperplane of κ. The orbit of (x_{1}, …, x_{m}) may likewise be characterized by the pair (G, W′) where W′ = ℝμ ∈ ℙ(κ) is defined by W μ^{T} = 0. For all λ ∈ ℝ^{m},

∑_{i=1}^{m} λ_{i}x_{i} = 0 ⇔ λG = 0 and λμ^{T} = 0  (5’)

Let t ∈ V be a fixed timelike vector. The set of points of (n − 1)-dimensional hyperbolic geometry in Weierstraß’s model is (cf. e.g. [1])

W^{n−1} := {x ∈ V | (x, x) = −1 ∧ (x, t) < 0}.

A hyperbolic motion is the restriction f|_{W^{n−1}} of an isometry f of (V, (·, ·)) which is orthochronous, (t, f(t)) < 0. The hyperbolic distance between x, y ∈ W^{n−1} is d(x, y) := arch(−(x, y)). Let x_{1}, …, x_{m} be m hyperbolic points. Their Gram matrix G = (g_{ij}) = −(ch d(x_{i}, x_{j})) satisfies

G = G^{T} , G has exactly one negative eigenvalue,

rank G ≤ n, g_{ij} ≤ −1 = g_{ii} for all i, j = 1, …, m. (9)

1If (V, (·, ·)) has signature (k, n − k), then the orbits of (m + 1)-tuples of vectors in the corresponding inhomogeneous orthogonal geometry are obtained analogously, by Theorem 1 and 2.

If x′_{1}, …, x′_{m} ∈ W^{n−1} have the same Gram matrix G, then there is an isometry of (V, (·, ·)) which takes x_{i} to x′_{i}, i = 1, …, m and which is orthochronous. On the other hand, let G be a real m × m matrix with (9) and let W := {λ ∈ ℝ^{m} | λG = 0}. From Theorem 2 there are x_{1}, …, x_{m} ∈ V with (4) and (5). Without loss of generality we may assume that (x_{1}, t) < 0. Then (x_{1}, x_{i}) < 0 implies (x_{i}, t) < 0 and x_{i} ∈ W^{n−1}, i = 1, …, m. Hence the orbits of m-tuples of points correspond to the m × m-matrices G which satisfy (9).

The set of points of (n − 1)-dimensional de-Sitter geometry is

S^{n−1} := {x ∈ V | (x, x) = 1}.

The motions are restrictions of the isometries of (V, (·, ·)) to S^{n−1}. Let x_{1}, …, x_{m} be m points of S^{n−1} with Gram matrix G and subspace W < ℝ^{m} defined by (5). If the space X < V spanned by x_{1}, …, x_{m} is not in 𝔏, then W = {λ ∈ ℝ^{m} | λG = 0}, and the orbit of (x_{1}, …, x_{m}) corresponds to the matrix G with

G = G^{T} , g_{11} = … = g_{mm} = 1,

G has at most one negative and at most n − 1 positive eigenvalues.

If X ∈ 𝔏, then there is a uniquely determined projective point ℝμ ∈ ℙ(V) with (5’). The orbit of (x_{1}, …, x_{m}) corresponds to the pair (G, ℝμ) where

G = G^{T} , g_{11} = … = g_{mm} = 1,

G is positive semi-definite, rank G < n − 1, μG = 0.

References

[1] Benz, W.: Real Geometries. BI Wissenschaftsverlag, Mannheim Leipzig Wien Zürich 1994.

[2] Blaschke, W.: Vorlesungen über Differentialgeometrie III. Verlag von Julius Springer, Berlin 1929.

[3] Huppert, B.: Angewandte Lineare Algebra. Walter de Gruyter, Berlin NewYork 1990.

Received March 3, 1998