PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 67(81) (2000), 121–132

BALL MODELS IN HERMITIAN SPACES

Boris Rosenfeld

Communicated by Mileva Prvanović

Abstract. Ball models of complex, quaternionic, and octonionic Hermitian elliptic and hyperbolic spaces are considered. The maps of straight lines and real geodesics in these models are found.

1. Hermitian spaces

In this paper we consider Hermitian elliptic and hyperbolic spaces over three division algebras: the field C of complex numbers, the skew field H of quaternions and the alternative skew field O of octonions. Hermitian elliptic spaces CS̄^{n}, HS̄^{n}, and OS̄^{2} are metrized projective spaces CP^{n}, HP^{n}, and OP^{2} over these algebras (note that in the case of algebra O only 2-plane can exist). A metric in these spaces is determined by imaginary absolute Hermitian hyperquadrics with equation

(1)  (x, x) = ∑_{i} ∑_{j} x̄^{i}a_{ij}x^{j} = 0, a_{ij} = ā_{ji}, i, j = 0, 1,…, n,

which can be reduced to the form

(2)  (x, x) = ∑_{i} x̄^{i}x^{i} = 0, i = 0, 1,…, n.

The forms (x, x) in (1) and (2) are Hermitian forms. The distance ω between points X(x^{i}) and Y (y^{i}) in these spaces is determined by the formula

(3)  cos^{2} \frac{ω}{r} = \frac{(x, y)(y, x)}{(x, x)(y, y)},

where r is the curvature radius of the space. Hermitian hyperbolic spaces CH̄^{n}, HH̄^{n}, and OH̄^{2} over algebras C, H and O are domains in the spaces CP^{n}, HP^{n}, and OP^{2}, bounded by oval absolute Hermitian hyperquadrics with equation (1) which can be reduced to the form

1991 Mathematics Subject Classification. Primary 53A35.

(4)  (x, x) = −x̄^{0}x^{0} + ∑_{i} x̄^{i}x^{i} = 0, i = 1, 2,…, n.

The distance ω between points X(x^{i}) and Y (y^{i}) in these spaces is determined by the formula

(5)  cosh^{2} \frac{ω}{q} = \frac{(x, y)(y, x)}{(x, x)(y, y)},

where qi is the imaginary curvature radius of these spaces.

The spaces CS̄^{n} and CH̄^{n} were defined by Fubini [1] and Study [2]; these spaces were considered by Coolidge [3] and Cartan [4], the space CS̄^{2} was studied in detail by Goldman [5]. Cartan [6] defined also the space HS̄^{n}. The plane OS̄^{2} was studied by Borel [7] and Freudenthal [8]. For all Hermitian elliptic and hyperbolic spaces see the books of the author [9, pp. 620–654 and 679–686] and [10, pp. 219–278 and 333–340].

The spaces CS̄^{n} and CH̄^{n} are Riemannian manifolds V^{2n} whose groups of motions are compact and noncompact simple Lie groups of class A_{n}; these symmetric spaces are symmetric spaces AIII according to classification of Cartan [6]. The spaces HS̄^{n} and HH̄^{n} are Riemannian manifolds V^{4n} whose groups of motions are compact and noncompact simple Lie groups of class C_{n+1}; these symmetric spaces are symmetric spaces CII according to classification [6]. The planes OS̄^{2} and OH̄^{2} are Riemannian manifolds V^{16}, whose groups of motions are compact and noncompact simple Lie groups of class F_{4}; these symmetric spaces are symmetric spaces FII according to classification [6].

The straight lines of these spaces are isometric to spheres Σ^{2} of radius r/2 and imaginary radius qi/2 in the Euclidean space R^{3}, respectively in the pseudo-Euclidean space R_{1}^{3}, to hyperspheres Σ^{4} of the same radii in spaces R^{5} and R_{1}^{5} and to hyperspheres Σ^{8} of the same radii in spaces R^{9} and R_{1}^{9}. Therefore Riemannian curvature of these manifolds in 2-directions located in “normal real n-chains” x^{i} = x̄^{i} is equal to

K = 1/r^{2}, respectively K = −1/q^{2},

and in 2-directions located in straight lines is equal to

K = 4/r^{2}, respectively K = −4/q^{2}.

Since 2-directions located in straight lines are called holomorphic, these Riemannian manifolds are called “Riemannian manifolds of constant holomorphic curvature”.

Let us call geodesics in Riemannian manifolds V^{2n} isometric to CS̄^{n} and CH̄^{n}, respectively in Riemannian manifolds V^{4n} isometric to HS̄^{n} and HH̄^{n} and in Riemannian manifolds V^{16} isometric to OS̄^{2} and OH̄^{2}, “real geodesics in the Hermitian spaces CS̄^{n} and CH̄^{n}, respectively in HS̄^{n}, HH̄^{n}, OS̄^{2}, and OH̄^{2}”.

If X and Y are two points in these spaces, a real geodesic joining points X and Y is always located in the straight line XY, and, since these straight lines are isometric to spheres Σ^{2}, Σ^{4}, and Σ^{8}, real geodesics in Hermitian spaces CS̄^{n} and CH̄^{n}, respectively in HS̄^{n}, HH̄^{n}, OS̄^{2} and OH^{2} are great circumferences on these spheres.

If in the space CP^{n}, respectively in HP^{n} or OP^{2}, a hyperplane at infinity is singled, this space without a singled hyperplane is called the affine space CE^{n}, respectively HE^{n} or OE^{2}. If the equation of a hyperplane at infinity is x^{0} = 0, the points of an affine space are determined by affine coordinates

(6)  X^{i} = x^{i}(x^{0})^{−1}.

An affine space, if in its hyperplane at infinity the metric of space CS̄^{n−1}, respectively HS̄^{n−1} and OS̄^{1}, is determined, is called Hermitian Euclidean space CR̄^{n}, respectively HR̄^{n} and OR̄^{2}. The distance ω between points X(X^{i}) and Y (Y^{i}) in Hermitian Euclidean spaces is determined by the formula

(7)  ω^{2} = (Y − X, Y − X)

where the form (X, X) is the form (2). The Euclidean space CR̄^{n}, respectively HR̄^{n} and OR̄^{2} is isometric to the real Euclidean space R^{2n}, respectively to R^{4n} and R^{16}.

An affine space, if in its hyperplane at infinity the metric of space CH̄^{n−1}, respectively HS̄^{n−1} and OH̄^{1}, is determined, is called a Hermitian pseudo-Euclidean space CR̄_{1}^{n}, respectively HR̄_{1}^{n} and OR̄_{1}^{2}. The distance ω between points X(X^{i}) and Y (Y^{i}) in Hermitian pseudo-Euclidean spaces is determined by formula (7) where the form (X, X) is the form (4). The pseudo-Euclidean space CR̄_{1}^{n}, respectively HR̄_{1}^{2} is isometric to the real pseudo-Euclidean space R_{2}^{2n}, respectively to R_{4}^{4n} and R_{8}^{16}.

A complex Hermitian Euclidean space was defined by Study [2]; it was studied by Coolidge [3]. For Hermitian Euclidean and pseudo-Euclidean spaces see [9, pp. 568–578] and [10, pp. 168–199 and 338–339].

The ball model of space CH̄^{n} was considered already by Fubini [1] and Study [2], see also [5, pp. 47–81]. In this paper analogous models of all spaces CS̄^{n}, HS̄^{n}, OS̄^{2}, CH̄^{n}, HH̄^{n}, OH̄^{2} are considered.

2. Ball models of real spaces

The real elliptic space S^{n} can be defined as metrized real projective space P^{n} with metric determined by the formula (3) where (x, x) = 0 is the equation of an imaginary absolute hyperquadric. The real hyperbolic space H^{n} can be defined as a domain in P^{n} bounded by an oval absolute hyperquadric with metric determined by the formula (5) where (x, x) = 0 is the equation of an absolute hyperquadric.

If in the space P^{n} a hyperplane at infinity x^{0} = 0 is singled, coordinates (6) and a metric of Euclidean space R^{n} are introduced, imaginary and oval absolute hyperquadrics (2) and (4) can be regarded as imaginary hypersphere CΣ^{n−1} with equation

(8)  (X, X) = −1

and real hypersphere Σ^{n−1} with equation

(9)  (X, X) = 1

in the space R^{n}. Therefore the space S^{n} can be interpreted in the space R^{n} with a hyperplane at infinity and imaginary hypersphere (8), and the space H^{n} can be interpreted in a ball B^{n} in the space R^{n} bounded by the hypersphere (9).

The space S^{n} with curvature radius r, respectively the space H^{n} with curvature radius qi, can be also defined as the hypersphere

(10)  (X, X) = r^{2}

of radius r with identified antipodal points in Euclidean space R^{n+1}, respectively as the hypersphere

(11)  (X, X) = −q^{2}

of imaginary radius qi in pseudo-Euclidean space R_{1}^{n+1}, also with identified antipodal points. Projective interpretations of spaces S^{n} and H^{n} in the space R^{n} can be also obtained by projecting hyperspheres (10) and (11) from their centers onto tangent hyperplanes X^{n+1} = r and X^{n+1} = q to these hyperspheres. These hyperplanes are spaces R^{n} and intersections of these hyperplanes with imaginary and real hypercones (X, X) = 0 are, respectively, the imaginary sphere CΣ^{n−1} and the real sphere Σ^{n−1}.

If we project the hyperspheres (10) and (11) from their poles with coordinates 0,…, 0, −r and 0,…, 0, −q onto the diametral hyperplane X^{n+1} = 0, we obtain conformal interpretations of S^{n} and H^{n} in R^{n}. These projections are stereographic projections and preserve angles between curves and map straight lines and circumferences in S^{n} and H^{n} onto straight lines or circumferences in R^{n}. The intersection of the hyperplane X^{n+1} = 0 with the hypersphere (10), respectively (11), is the real equatorial sphere Ω^{n−1}, respectively the imaginary sphere CΩ^{n−1}. A stereographic projection of the hypersphere (10) maps its upper hemisphere into the ball B^{n} bounded by the real sphere Ω^{n−1} and its lower hemisphere onto the exterior domain of this ball. The stereographic projection of the hypersphere (11), consisting of two separate hemispheres, maps its upper hemisphere into the ball B^{n} bounded by the real sphere Σ^{n−1} and its lower hemisphere onto the exterior domain of this ball. The sphere Σ^{n−1} is cut from the hyperplane X^{n+1} = 0 by a hypercone with rectilinear generators parallel to rectilinear generators of the asymptotic hypercone of the hypersphere (11).

Since straight lines in spaces S^{n} and H^{n} are mapped on the hyperspheres (10) and (11) into their sections by diametral 2-planes and these 2-planes cut from the hyperplanes X^{n+1} = r and X^{n+1} = q straight lines, in the projective interpretation of S^{n}, respectively of H^{n}, in R^{n} straight lines are mapped into straight lines, respectively by chords in ball B^{n}.

Since under stereographic projection circumferences on hyperspheres (10) and (11) are mapped in the hyperplanes X^{n+1} = 0 into circumferences or straight lines, in conformal interpretations of S^{n} and H^{n} in R^{n} straight lines are mapped into circumferences or straight lines or by their arcs and segments. Since great circles on hypersphere (10) meet its equatorial great sphere Ω^{n−1} at two antipodal points, in the conformal interpretation of S^{n} in R^{n} straight lines are mapped into diameters of the sphere Ω^{n−1} and circumferences meeting this sphere at two antipodal points.

Since sections of the hypersphere (11) by its diametral 2-planes go into themselves under reflection of this hypersphere from its center, and under stereographic projection this reflection is imaged on the hyperplane X^{n+1} = 0 by inversion in the sphere Σ^{n−1} since it is the intersection of this hyperplane with a hypercone whose rectilinear generators are parallel to the rectilinear generators of the asymptotic hypercone of the hypersphere (11), the straight lines and circumferences going into themselves under inversion in the sphere are orthogonal to this sphere. Therefore in the conformal interpretation of H^{n} in R^{n} straight lines are imaged by diameters of ball B^{n} and arcs of circumferences in this ball orthogonal to its boundary Σ^{n−1}.

If two points X and Y in S^{n} and H^{n} are mapped in projective interpretations of these spaces in R^{n} into points X′ and Y′, the straight line XY is mapped into the straight line X′Y′. If two points X and Y in H^{n} are mapped in the conformal interpretation of this space in R^{n} into points X′ and Y′, the straight line XY is mapped into the are of circumference through points X′, Y′, and the points X′′ and Y′′ obtained from X′ and Y′ by inversion in the sphere Σ^{n−1}. If two points X and Y in S^{n} are mapped in the conformal interpretation of this space in R^{n} into points X′ and Y′, the straight line XY is mapped into the circumference through points X′ and Y′, and the points X′′ and Y′′ obtained from X′ and Y′ by inversion in the sphere Ω^{n−1} and by reflection from its center; the product of this inversion and reflection is the inversion in the imaginary sphere CΣ^{n−1} with equation (8).

The models of H^{n} obtained by its projective and conformal interpretations in a ball B^{n} are called “ball models of H^{n}”. Ball models of H^{n} are multidimensional generalizations of disk models of H^{2} obtained by classical Beltrami-Klein and Poincaré interpretations of this plane. We will call the model of S^{n} in R^{n} with a ball B^{n} bounded by the sphere Ω^{n−1} “ball model of space S^{n}”.

Let triangles ABC in planes S^{2} and H^{2} be mapped in disk models of these planes into curvilinear triangles A′B′C′ whose one side B′C′ is a segment of the diameter of the disk B^{2} and the sides A′B′ and A′C′ are arcs of circumferences which in the case of S^{2} meet the boundary of B^{2} at two antipodal points and in the case of H^{2} are orthogonal to the boundary of B^{2}. Comparison of curvilinear triangles A′B′C′ with rectilinear triangles A′B′C′ clearly shows that the sum of angles in curvilinear triangles A′B′C′ and consequently in the triangle ABC in S^{2} are > 2π, and in H^{2} are < 2π.

3. Ball models of Hermitian spaces

The Hermitian elliptic spaces CS̄^{n}, HS̄^{n}, and OS̄^{2} and the Hermitian hyperbolic spaces CH̄^{n}, HH̄^{n}, and OH̄^{2} also admit projective interpretations in the Hermitian Euclidean spaces CR̄^{n}, HR̄^{n}, and OR̄^{2}, each space in a Euclidean space over the same algebra. The space CS̄^{n}, respectively HS̄^{n} and OS̄^{2}, can be interpreted in the space CR̄^{n}, respectively HR̄^{n} and OR̄^{2}, with affine coordinates (6) of points and imaginary Hermitian hypersphere (8). The space CH̄^{n}, respectively HH̄^{n} and OH̄^{2}, can analogously be interpreted in the ball CB̄^{n} bounded by the Hermitian hypersphere (9) in CR̄^{n}, respectively in the balls HB̄^{n} and OB̄^{2} bounded by Hermitian hyperspheres (9) in HR̄^{n} and OR̄^{2}.

Since the Hermitian Euclidean space CR̄^{n}, respectively HR̄^{n} and OR̄^{2}, is isometric to the real Euclidean space R^{2n}, respectively in R^{4n} and R^{16}, the space CS̄^{n}, respectively HS̄^{n} and OS̄^{2}, and the space CH̄^{n}, respectively HH̄^{n} and OH̄^{2}, admit interpretations in the real Euclidean space R^{2n}, respectively in R^{4n} and R^{16}. In these interpretations the Hermitian hypersphere (9) in CR̄^{n}, respectively in HR̄^{n} and OR̄^{2}, is imaged by the hypersphere Σ^{2n−1}, respectively by Σ^{4n−1} and Σ^{15}, and the imaginary Hermitian hypersphere (8) in CR̄^{n}, respectively HR̄^{n} and OR̄^{2}, is imaged by the imaginary hypersphere CΣ^{2n−1}, respectively by CΣ^{4n−1} and CΣ^{15}.

The model of CH̄^{n}, respectively of HH̄^{n} and OH̄^{2}, obtained by its projective interpretations in a ball B^{2n}, respectively in B^{4n} and B^{16} is called “ball models of CH̄^{n}, respectively of HH̄^{n} and OH̄^{2}”. We will call the model of CS̄^{n}, respectively of HS̄^{n} and OS̄^{2}, obtained by its projective interpretation in R^{2n} with a ball B^{2n} bounded by sphere Ω^{2n−1}, respectively in R^{4n} and R^{16} with balls B^{4n} and B^{16} bounded by spheres Ω^{4n−1} and Ω^{15} “ball model of CS̄^{n}, respectively of HS̄^{n} and OS̄^{2}”.

4. Maps of straight lines and real geodesics

Theorem 1. In a ball model of the space CS̄^{n}, respectively of HS̄^{n} and OS̄^{2}, straight lines are imaged by 2-planes in R^{2n} with stereographic projections of 2spheres of radius r/2, respectively by 4-planes in R^{4n} and 8-planes in R^{16} with stereographic projections of 4-spheres and 8-spheres of the same radius.

Proof. Straight lines in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, are imaged by 2-planes, respectively 4-planes and 8-planes, because a straight line in CS̄^{n}, respectively HS̄^{n} and OS̄^{2}, is the space CS̄^{1}, respectively HS̄^{1} and OS̄^{1}.

For the space CS̄^{n} consider the sphere (X^{0})^{2} + (X^{1}) + (X^{2})^{2} = 1 in R^{3} and its stereographic projection from its pole with coordinates 1, 0, 0 onto the plane X^{0} = 0. If we consider the plane X^{0} = 0 as the plane of complex variable x, and if a point X of the sphere with coordinates X^{0}, X^{1}, X^{2} is projected onto a point x in the complex plane, the complex number x = x_{1} +ix_{2} is connected with coordinates X^{0}, X^{1}, X^{2} by correlations

(12)  X^{0} = \frac{x̄x − 1}{x̄x + 1},  X^{1} = \frac{2x_{1}}{x̄x + 1},  X^{2} = \frac{2x_{2}}{x̄x + 1}

Since x_{1} = \frac{x+x̄}{2} and x_{2} = \frac{x−x̄}{2i}, the inner product

cos ω = XY = X^{0}Y^{0} + X^{1}Y^{1} + X^{2}Y^{2}

is equal to

cos ω = \frac{(x̄x − 1)(ȳy − 1) + (x + x̄)(y − ȳ) − (x − x̄)(y − ȳ)}{(x̄x + 1)(ȳy + 1)}.

Therefore

(13)  cos^{2} \frac{ω}{2} = \frac{1 + cos ω}{2} = \frac{(x̄y + 1)(ȳx + 1)}{(x̄x+!)(ȳy + 1)}.

But the right-hand side of (13) coincides with the right hand side of (3) for n = 1, x^{0} = y^{0} = 1, x^{1} = x, y^{1} = y, that is ω can be regarded as a distance between two points in the line CS̄^{1} with curvature radius 2. But ω is a distance on the sphere of radius 1. Therefore straight lines in CS̄^{n} with curvature radius r are isometric to spheres Σ^{2} of radius r/2 and their images in the ball model are 2-planes with stereographic projections of 2-spheres.

For the space HS̄^{n} consider the hypersphere (X^{0})^{2} + (X^{1})^{2} + (X^{2})^{2} + (X^{3})^{2} + (X^{4})^{2} = 1 in R^{5} and its stereographic projection from its pole with coordinates 1, 0, 0, 0, 0 onto the hyperplane X^{0} = 0. If we consider the plane X^{0} = 0 as a space of a quaternion variable x and if the projection of a point X of the hypersphere with coordinates X^{0}, X^{1}, X^{2}, X^{3}, X^{4} is projected onto the point x in the quaternionic space, the quaternion x = x_{1} + ix_{2} + jx_{3} + kx_{4} is related to the coordinates X^{0}, X^{1}, X^{2}, X^{3}, X^{4} by correlations (12). Since x_{1} = (x + x̄)/2, x_{2} = (ix̄ − xi)/2, x_{3} = (jx̄ − xj)/2, x_{4} = (kx̄ − xk)/2, the inner product

cos ω = XY = X^{0}Y^{0} + X^{1}Y^{1} + X^{2}Y^{2} + X^{3}Y^{3} + X^{4}Y^{4}

is equal to

cos ω =

\frac{(x̄x−1)(ȳy−1)+(x+x̄)(y+ȳ)+(ix̄−xi)(iȳ−yi)+(jx̄−xj)(jȳ−yj)+(kx̄−xk)(kȳ−yk)}{(x̄x+1)(ȳy+1)}.

Hence we obtain (13). But the right-hand side of (13) coincides with the right hand side of (3) for n = 1, x^{0} = y^{0} = 1, x^{1} = x, y^{1} = y, that is ω can be regarded as a distance between two points on the line CS̄^{1} with curvature radius 2. But ω was a distance on the hypersphere of radius 1. Therefore straight lines in HS̄^{n} with curvature radius r are isometric to 4-spheres Σ^{4} of radius r/2 and their images in the ball model are 4-planes with stereographic projections of 4-spheres.

The proof for the plane OS̄^{2} is analogous.

The 2-planes, respectively 4-planes and 8-planes, imaging straight lines in the space CS̄^{n}, respectively HS̄^{n} and OS̄^{2}, cut from imaginary hypersphere CΣ^{2n−1}, respectively CΣ^{4n−1} and CΣ^{15}, imaging absolute Hermitian hyperquadric of CS̄^{n}, respectively of HS̄^{n} and OS̄^{2}, imaginary circumferences CΣ^{1}, respectively imaginary 3-spheres CΣ^{3} and 7-spheres CΣ^{7}.

An inversion in the imaginary hypersphere CΣ^{2n−1}, respectively CΣ^{4n−1} and CΣ^{15}, is a product of a reflection from center of this imaginary hypersphere and an inversion in the real hypersphere Ω^{2n−1}, respectively in the real hyperspheres Ω^{4n−1} and Ω^{15}, with the same center. The 2-planes, respectively 4-planes and 8-planes, imaging straight lines cut from hypersphere Ω^{2n−1}, respectively from hyperspheres Ω^{4n−1} and Ω^{15}, circumferences Ω^{1}, respectively, 3-spheres Ω^{3} and 7-spheres Ω^{7} which are boundaries of disks B^{2}, respectively of balls B^{4} and B^{8}.

The interpretation of S^{n} as a hypersphere in R^{n+1} with identified antipodal points is generalized for CS̄^{n}, respectively for HS̄^{n} and OS̄^{2}, as follows: a Hermitian hypersphere of radius r in CR̄^{n+1}, respectively in HR̄^{n+1} and OR̄^{3}, is interpreted in R^{2n+2}, respectively in R^{4n+4} and R^{24} by the hypersphere Σ^{2n+1}, respectively by Σ^{4n+3} and Σ^{23} of the same radius. Points in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, of curvature 1/r^{2} can be represented by straight lines in CR̄^{n+1}, respectively in HR̄^{n+1} and OR̄^{3}, through the center of a Hermitian hypersphere and by 2-planes in R^{2n+2}, respectively by 4-planes in R^{4n+4} and 8-planes in R^{24}, through the center of Σ^{2n+1}. These 2-planes, respectively 4-planes and 8-planes, cut from Σ^{2n+1}, respectively from Σ^{4n+3} and Σ^{23}, great circumferences, respectively 3-spheres and 7-spheres, of a congruence. If we identify antipodal points in Σ^{2n+1}, respectively in Σ^{4n+3} and Σ^{23}, we obtain the space S^{2n+1}, respectively spaces S^{4n+3} and S^{23}, and the circumferences, respectively 3-spheres and 7-spheres, of this congruence become paratactic straight lines in S^{2n+1}, respectively 3-planes in S^{4n+3} and 7-planes in S^{23}. Therefore the congruence of circumferences, respectively of 3-spheres and 7-spheres, is called paratactic congruence and points in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, are represented by circumferences, respectively by 3-spheres and 7-spheres, of this congruence. Each circumference, respectively 3-sphere and 7-sphere, of this congruence meets 2n-plane in R^{2n+2}, respectively 4n-plane in R^{4n+4} and R^{24}, imaging a hyperplane in CR̄^{n+1}, respectively in HR̄^{n+1} and OR̄^{3}, at a single point. These points are maps of points of CS̄^{n} in R^{2n}, respectively of HS̄^{n} in R^{4n} and of OS̄^{2} in R^{16}.

Theorem 2. Images of straight lines in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, can be obtained by projecting the hypersphere Σ^{2n+1}, respectively Σ^{4n+3} and Σ^{23}, from its center onto a 2n-plane, respectively onto a 4n-plane and a 16-plane.

Since straight lines CS̄^{1}, respectively HS̄^{1} and OS̄^{1}, are isometric to the sphere Σ^{2}, respectively to the 4-sphere Σ^{4} and 8-sphere Σ^{8}, of radius r/2, we can obtain a projection of such a line from the center of a Hermitian hypersphere onto a 2plane in R^{2n}, respectively onto a 4-plane in R^{4n} and an 8-plane in R^{16}, imaging this line; if we consider 3-plane, respectively 5-plane and 9-plane, containing this 2-plane, respectively 4-plane and 8-plane, and the center of the hypersphere, and in this 3-plane, respectively 5-plane and 9-plane, the 2-sphere, respectively 4-sphere and 8-sphere, of radius r/2 tangent to the real hypersphere imaging the Hermitian hypersphere and passing through the center of the real hypersphere. Projecting 2planes, respectively 4-planes and 8-planes, meet this 2-sphere, respectively 4-sphere and 8-sphere, also at a single point, and a projection of a line in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, onto 2-plane, respectively 4-plane and 8-plane, coincides with stereographic projection of sphere Σ^{2}, respectively 4-sphere Σ^{4} and 8-sphere Σ^{8} of radius r/2 from its pole.

Theorem 3. In ball models of the space CH̄^{n}, respectively of HH̄^{n} and OH̄^{2} straight lines are mapped into by disks B^{2} on 2-planes in R^{2n} with stereographic projections of 2-spheres of radius qi/2, respectively into balls B^{4} in 4-planes in R^{4n} and balls B^{8} in 8-planes in R^{16} with stereographic projections of 4-spheres and 8-spheres of the same radius.

Proof. Straight lines in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, are mapped into disks B^{2} in 2-planes, respectively by balls B^{4} in 4-planes and B^{8} in 8-planes, because a straight line in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, is the space CH̄^{1}, respectively HH̄^{1} and OH̄^{1}. The boundaries Σ^{1} of disks B^{2}, respectively boundaries Σ^{3} of balls B^{4} and boundaries Σ^{7} of balls B^{8} are intersections of 2-planes of disks B^{2}, respectively of 4-planes of balls B^{4} and 8-planes of balls B^{8} with hypersphere Σ^{2n−1}, respectively Σ^{4n−1} and Σ^{15}, imaging absolute Hermitian hyperquadric of space CH̄^{n}, respectively HS̄^{n} and OH̄^{2}.

For the space CH̄^{n} consider the sphere −(X^{0})^{2} + (X^{1})^{2} + (X^{2})^{2} = −1 in R_{1}^{3} and its stereographic projection from its pole with coordinates 1, 0, 0 onto the plane X^{0} = 0. If we regard the plane X^{0} = 0 as a plane of complex variable x and if a point X of the sphere with coordinates X^{0}, X^{1}, X^{2} is projected onto a point x in complex plane, the complex number x = x_{1} +ix_{2} is connected with the coordinates X^{0}, X^{1}, X^{2} by the relations

(14)  X^{0} = \frac{x̄x + 1}{x̄x − 1},  X^{1} = \frac{2x_{1}}{x̄x − 1},  X^{2} = \frac{2x_{2}}{x̄x − 1}

Since x_{1} = (x + x̄)/2 and x_{2} = (x − x̄)/2i, the inner product cosh ω = XY = −X^{0}Y^{0} + X^{1}Y^{1} + X^{2}Y^{2} is equal to

cosh ω = \frac{−(x̄x + 1)(ȳy + 1) + (x + x̄)(y + ȳ) − (x − x̄)(y − ȳ)}{(x̄x − 1)(ȳy − 1)}.

Therefore

(15)  cosh^{2} \frac{ω}{2} = \frac{1 + cosh ω}{2} = \frac{(x̄y − 1)(ȳx − 1)}{(x̄x − 1)(ȳy − 1)}.

But the right-hand side of (15) coincides with the right-hand side of (5) for n = 1, x^{0} = y^{0} = 1, x^{1} = x, y^{1} = y, that is ω/2 can be regarded as a distance between two points on the line CH̄^{1} of curvature radius 2i. But ω is a distance on the sphere of imaginary radius i. Therefore straight lines in CH̄^{n} with curvature radius qi are isometric to spheres Σ_{1}^{2} of radius qi/2 and their images in the ball model are 2-planes with stereographic projections of 2-spheres of imaginary radius.

For the space HH̄^{n} consider the hypersphere −(X^{0})^{2}+(X^{1})^{2}+(X^{2})^{2}+(X^{3})^{2}+ (X^{4})^{2} = −1 in R_{1}^{5} and its stereographic projection from its pole with coordinates 1, 0, 0, 0, 0 onto the hyperplane X^{0} = 0. If we regard the hyperplane X^{0} = 0 as a space of quaternion variable x and if a point X of the hypersphere with coordinates X^{0}, X^{1}, X^{2}, X^{3}, X^{4} is projected onto a point x in quaternion space, the quaternion x = x_{1} + ix_{2} + jx_{3} + kx_{4} is related to the coordinates X^{0}, X^{1}, X^{2}, X^{3}, X^{4} by relations (14). Since x_{1} = (x+x̄)/2, x_{2} = (ix̄−xi)/2, x_{3} = (jx̄−xj)/2, x_{4} = (kx̄− xk)/2, the inner product cosh ω = XY = −X^{0}Y^{0} +X^{1}Y^{1} +X^{2}Y^{2} +X^{3}Y^{3} +X^{4}Y^{4} is equal to

cosh ω =

\frac{−(x̄x+1)(ȳy+1)+(x+x̄)(y+ȳ)+(ix̄−xi)(iȳ−yi)+(jx̄−xj)(jȳ−yj)+(kx̄−xk)(kȳ−yk)}{(x̄x−1)(ȳy−1)}.

Hence we obtain (15). But the right hand side of (15) coincides with the right hand side of (5) for n = 1, x^{0} = y^{0} = 1, x^{1} = x, y^{1} = y, that is ω/2 can be regarded as a distance between two points on the line CH̄^{1} with curvature radius 2i. But ω is a distance on the hypersphere of radius i. Therefore straight lines in HH̄^{n} with curvature radius qi are isometric to 4-spheres Σ^{4} of radius qi/2 and their images in the ball model are 4-planes with stereographic projections of 4-spheres of imaginary radius.

The proof for the plane OH̄^{2} is analogous.

Theorem 4. Images of straight lines in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, can be obtained by projecting the hypersphere Σ_{1}^{2n+1}, respectively Σ_{1}^{4n+3} and Σ_{1}^{23}, from its center onto a 2n-plane, respectively onto a 4n-plane and a 16-plane.

Since straight lines CH̄^{1}, respectively HH̄^{1} and OH̄^{2}, are isometric to the sphere Σ_{1}^{2}, respectively to the 4-sphere Σ_{1}^{4} and 8-sphere Σ_{1}^{8}, of imaginary radius qi/2, we can obtain a projection of such a line from the center of the Hermitian hypersphere onto a 2-plane in R^{2n}, respectively onto a 4-plane in R^{4n} and an 8-plane in R^{16}, imaging this line; we consider a 3-plane, respectively a 5-plane and a 9-plane, containing this 2-plane, respectively 4-plane and 8-plane, and the center of a hypersphere, and in this 3-plane, respectively 5-plane and 9-plane, the 2-sphere, respectively 4-sphere and 8-sphere, of radius qi/2 tangent to the real hypersphere imaging the Hermitian hypersphere and passing through the center of a real hypersphere. The projecting 2-planes, respectively 4-planes and 8-planes, meet this 2-sphere, respectively 4-sphere and 8-sphere, also at a single point, and the projection of a line in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, onto a 2-plane, respectively 4-plane and 8-plane, coincides with the stereographic projection of the sphere Σ_{1}^{2}, respectively the 4-sphere Σ_{1}^{4} and the 8-sphere Σ_{1}^{8} of radius qi/2 from its pole.

Since a real geodesic joining points X and Y in Hermitian spaces CS̄^{n} and CH̄^{n}, respectively in HS̄^{n}, HH̄^{n}, OS̄^{2}, and OH̄^{2}, is always located on the straight line XY, the study of real geodesics in these spaces can be reduced to the study of real geodesics in straight lines.

Theorem 5. In a ball model of the space CS̄^{n}, respectively of HS̄^{n} and OS̄^{2}, real geodesics are mapped into by diameters of disks B^{2} in 2-planes imaging straight lines in CS̄^{n}, respectively of balls B^{4} and B^{8} in 4-planes and 8-planes imaging straight lines in HS̄^{n} and OS̄^{2}, and by circumferences in these 2-planes, respectively in these 4-planes and 8-planes, intersecting boundaries of disks B^{2}, respectively of balls B^{4} and B^{8}, in two antipodal points. If two points X and Y in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, are located on a diameter of a disk B^{2}, respectively of a ball B^{4} or B^{8}, a real geodesic joining X and Y is mapped into straight line coinciding with this diameter. If two points X and Y in CS̄^{n}, respectively in HS̄^{n} and OS̄^{2}, are not located on a diameter of a disk B^{2}, respectively of a ball B^{4} or B^{8}, a real geodesic joining X and Y is mapped into a circumference through points X′ and Y′ imaging X and Y, and points X′′ and Y′′ obtained from X′ and Y′ by inversion in the boundary of this disk, respectively of these balls and by reflection from the center of disk, respectively of balls.

The theorem is a consequence of Theorem 1 and the properties of ball models of the space S^{n}.

Theorem 6. In ball models of space CH̄^{n}, respectively of HH̄^{n} and OH̄^{2}, real geodesics are mapped into diameters of disks B^{2} in 2-planes imaging straight lines in CH̄^{n}, respectively of balls B^{4} and B^{8} in 4-planes and 8-planes imaging straight lines in HH̄^{n} and OH̄^{2}, and by arcs of circumferences orthogonal to boundaries of these disks, respectively of balls. If two points X and Y in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, are located on a diameter of a disk B^{2}, respectively of ball B^{4} or B^{8}, a real geodesic joining X and Y is mapped into this diameter. If two points X and Y in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, are not located on a diameter of a disk B^{2}, respectively of a ball B^{4} or B^{8}, a real geodesic joining X and Y is mapped into arc of circumference through points X′ and Y′ imaging X and Y, and points X′′ and Y′′ in 2-plane of the disk, respectively in 4-plane or 8-plane of the balls, obtained from X′ and Y′ by inversion in the boundary of this disk, respectively of these balls.

The theorem is a consequence of Theorem 3 and the properties of ball models of the space H^{n}.

The case when points X and Y in CH̄^{n}, respectively in HH̄^{n} and OH̄^{2}, are points of an absolute Hermitian hyperquadric of this space can be obtained from the general case by a limiting process when the points X′ and X′′ tend to one another, as well as Y′ and Y′′. In the most simple case when X and Y are points in CH̄^{1} a real geodesic joining X and Y is imaged by an arc of circumference through points X′ and Y′ imaging X and Y whose center C is the pole of straight lines X′Y′: since this circumference is orthogonal to the boundary of the disk, the radii CX′ and CY′ of this circumference are tangent to the boundary of the disk and C is the meeting point of two tangents at X′ and Y′, that C is the pole of the straight line X′Y′.

Goldman [5, p. 62–63] imaged a map of a straight line in the ball model of CH̄^{n} coinciding with the Poincaré interpretation of H^{2} in a disk in R^{2n}, as well as a map of a 2-plane in a normal real n-chain in CH̄^{n} coinciding with the Beltrami-Klein interpretation of H^{2} in a disk in R^{2}. In these maps he imaged coordinate lines of Fermi coordinates in H^{2}, that is a straight line, its equidistant conics, and straight lines orthogonal to the first straight line and to its equidistant conics. In [5, p. 77] Goldman gave the equation of an image in R^{2n} of a real geodesic in CH̄^{n} joining points X and Y of an absolute Hermitian hyperquadric of this space. These maps and equations are reproduced in [10, p. 235].

References

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[2] E. Study, Kürzeste Wege im komplexen Gebiete, Math. Annalen 60 (1905), 321–377.

[3] J.L. Coolidge, The Geometry of the Complex Domain, Clarendon Press, Oxford, 1924.

[4] É. Cartan, Leçons sur la géométrie projective complexe, Gauthier-Villars, Paris, 1950.

[5] W.M. Goldman, Complex Hyperbolic Geometry, Maryland University Press, 1995.

[6] É. Cartan, Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple, in: Oeuvres complètes, vol. 1, CRNS, Paris, 1984, pp. 867–989.

[7] A. Borel, Le plan projective des octaves et les sphères comme les espaces homogènes, in: Collected Papers, vol. 1, Springer-Verlag, Berlin, New York, 1983, pp. 39–41.

[8] H. Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, Geometria Dedicata 19 (1985), 1–63.

[9] B.A. Rosenfeld, Neevklidovy geometrii, Gostehizdat, Moskva, 1955.

[10] B.A. Rosenfeld, Geometry of Lie Groups, Kluwer, Dordrecht-Boston-London, 1997.

Department of Mathematics (Received 26 06 1998) Pennsylvania State University University Park, PA 16802 USA