数 学 杂 志 Vol. 34 ( 2014 ) J. of Math. (PRC) No. 2 THE LAW OF SINES FOR AN n-SIMPLEX IN HYPERBOLIC SPACE AND SPHERICAL SPACE AND ITS APPLICATIONS WANG Wen, YANG Shi-guo, YU Jing, Qi Ji-bing (Department of Mathematics, Hefei Normal University, Hefei 230601, China) Abstract: In this paper, the law of sines and related geometric inequalities for an n-simplex in an n-dimensional hyperbolic space and an n-dimensional spherical space are studied. By using the theory and method of distance geometry, we give the law of sines for an n-simplex in an n- dimensional hyperbolic space and an n-dimensional spherical space. As applications, we obtain Hadamard type inequalities and Veljan-Korchmaros type inequalities in n-dimensional hyperbolic space and n-dimensional spherical space. In addition, some new geometric inequality about“metric addition”involving volume and n-dimensional space angle of simplex in Hn(K) and Sn(K) is established. Keywords: hyperbolic space; spherical space; the law of sine; metric addition; geometric inequality 2010 MR Subject Classi¯cation: 51K05; 52A40; 52A55 Document code: A Article ID: 0255-7797(2014)02-0214-11 1 Introduction The law of sine of triangle (4ABC) in the Euclidean plane is well known as follows a b c abc = = = ; (1.1) sin A sin B sin C 2S È 1 where S = p(p ¡ a)(p ¡ b)(p ¡ c), p = 2 (a + b + c). Let a; b; c be the edge-lengths of a triangle ABC in the hyperbolic space with curvature ¡1. Then we have the law of sine of hyperbolic triangle ABC as follows (see [1]) sinh a sinh b sinh c sinh a sinh b sinh c = = = ; (1.2) sin A sin B sin C 2¢ È 1 where ¢ = sinh p(sinh p ¡ sinh a)(sinh p ¡ sinh b)(sinh p ¡ sinh c), p = 2 (a + b + c). ¤ Received date: 2012-05-18 Accepted date: 2012-11-05 Foundation item: Supported by the Doctoral Programs Foundation of Education Ministry of China (20113401110009); Natural Science Research Project of Hefei Normal University (2012kj11); Uni- versities Natural Science Foundation of Anhui Province (KJ2013A220; KJ2011B133). Biography: Wang Wen(1985{), male, born at Zongyang, Anhui, master, major in convex and distance geometry. No. 2 The law of Sines for an n-simplex in hyperbolic space and spherical space ¢ ¢ ¢ 215 We denote by a; b; c the edge-lengths of a triangle ABC in the spherical space with curvature 1. Then we have the law of sine of spherical triangle ABC as follows (see [2]) sin a sin b sin c sin a sin b sin c = = = ; (1.3) sin A sin B sin C 2¢ È 1 where ¢ = sin p(sin p ¡ sin a)(sin p ¡ sin b)(sin p ¡ sin c), p = 2 (a + b + c) 2 (0; ¼). The law of sine of triangle in Euclidean plane were generalized to the n-dimensional n simplex in n-dimensional Euclidean space E . Let fA0;A1; ¢ ¢ ¢ ;Ang be the vertex sets of n n-dimensional simplex ­n(E) in the n-dimensional Euclidean E . Denote by V the volume of the simplex ­n(E), and Fi (i = 0; 1; ¢ ¢ ¢ ; n) the areas of i-th face fi = fA0;A1; ¢ ¢ ¢ ; Ai¡1;Ai+1; ¢ ¢ ¢ ;Ang ((n ¡ 1)-dimensional simplex) of the simplex ­n(E). In 1968, Bators de¯ned the n-dimensional sines of the n-dimensional vertex angles ®i (i = 0; 1; ¢ ¢ ¢ ; n) for the n-dimensional simplex ­n(E), and established the law of sines for ­n(E) as follows (see [3]) Q n F0 F1 (n ¡ 1)! j=0 Fj = = ¢ ¢ ¢ = n¡1 : (1.4) sin ®0 sin ®1 (nV ) Obviously, formula (1:4) is generalization of formula (1:1) in n-dimensional Euclidean space En. Then, some di®erent forms of generalization about formula (1:1) was given in [4, 5, 6]. From 1970s, many geometry researchers were attempted to generalize formulas (1:2) and (1:3) to an n-dimensional hyperbolic simplex (spherical simplex), to establish the law of sines in n-dimensional hyperbolic space Hn and in n-dimensional spherical space Sn. In 1978, Erikson de¯ned the n-dimensional polar sine of i-th face fi(Pi 2= fi) of ­n(S) in Sn;1 (see [7]) as follows n Pol sin Fi = j[º0; º1; ¢ ¢ ¢ ; ºi¡1; ºi+1; ¢ ¢ ¢ ; ºn]j (i = 0; 1; ¢ ¢ ¢ ; n): (1.5) n Let sin Pi be the n-dimensional sine of the i-th angle of ­n(S) (see [7]). The law of sines in the n-dimensional spherical space Sn;1 was obtained in [7] as follows n n n Pol sin F0 Pol sin F1 Pol sin Fn n = n = ¢ ¢ ¢ = n : (1.6) sin P0 sin P1 sin Pn In 1980s, Yang and Zhang (see [8, 9, 10]) made a large number of basic works of geometric inequality in n-dimensional hyperbolic space Hn and in n-dimensional spherical space Sn, and established the law of cosines in n-dimensional hyperbolic space Hn and n-dimensional spherical space Sn. But they did not establish the law of sines in n-dimensional hyperbolic space Hn and n-dimensional spherical space Sn. In addition, some new geometric inequality about “metric addition”involving volume and n-dimensional space angle of simplex in Hn(K) and Sn(K) is established. In this paper, we give generalizations of (1:2) and (1:3) in n-dimensional hyperbolic space Hn and in n-dimensional spherical space Sn, and establish the law of sines n-dimensional simplex in n-dimensional hyperbolic space Hn and n-dimensional spherical space Sn. As 216 Journal of Mathematics Vol. 34 their applications, we obtain Veljan-Korchmaros type inequalities and Hadamard type in- equalities in n-dimensional hyperbolic space Hn and n-dimensional spherical space Sn. 2 The Law of Sine in Hyperbolic Space We consider the model of the hyperbolic space in Euclidean space (see [9]). Let B be a set whose elements x(x1; x2; ¢ ¢ ¢ ; xn) are in an n-dimensional vector space 2 2 2 and meet the following condition x1 + x2 + ¢ ¢ ¢ + xn < 1. Given a distance between x and y in the set B, denote by xy satisfying p 1 ¡ x y ¡ x y ¡ ¢ ¢ ¢ ¡ x y cosh ¡Kxy = È 1 1 2 È2 n n : 2 2 2 2 2 2 1 ¡ x1 ¡ x2 ¡ ¢ ¢ ¢ ¡ xn 1 ¡ y1 ¡ y2 ¡ ¢ ¢ ¢ ¡ yn Then the metric space with this distance in Rn+1 is called n-dimensional hyperbolic space with curvature K(< 0), denote by Hn(K). n Let §n(H) be an n-dimensional simplex in the n-dimensional hyperbolic space H (K), and fP0;P1; ¢ ¢ ¢ ;Png be its vertexes, aij (0 6 i; j 6 n) be its edge-length, V be its volume, respectively. To give the law of sines in n-dimensional hyperbolic space Hn(K), we give the following de¯nition. De¯nition 2.1 Suppose that §n(H) = fP0;P1; ¢ ¢ ¢ ;Png be an n-dimensional simplex n in n-dimensional hyperbolic space H (K). n edges P0Pi (i = 1; 2; ¢ ¢ ¢ ; n) with initial point Ó P0 form an n-dimensional space angle P0 of the simplex §n(H). Let i; j be the angle formed by two edges P0Pi and P0Pj. The sine of the n-dimensional space angle P0 of the simplex §n(H) is de¯ned as follows 1 sin P0 = (det Q0) 2 ; (2.1) where 2 3 1 cos i;Ó j 6 7 6 7 6 1 7 Q0 = 6 . 7 (i; j = 1; 2; ¢ ¢ ¢ ; n): 4 .. 5 cos i;Ó j 1 Similarly, we can de¯ne the sine of the n-dimensional space angle Pi (i = 1; 2; ¢ ¢ ¢ ; n) of the simplex §n(H). At ¯rst, we prove that this de¯nition is sensible. Actually, for n-ray P0Pi (i = 1; 2; ¢ ¢ ¢ ; n) of the simplex §n(H) in n-dimensional hyper- bolic space Hn(K), and denote by i;Ó j (i; j = 1; 2; ¢ ¢ ¢ ; n) the included angle between two rays 0 0 P0Pi and P0Pj. According to [12], we know that there exist n-ray P0Pi (i = 1; 2; ¢ ¢ ¢ ; n) which are independence in n-dimensional Euclidean space En, such that the included angle between 0 0 0 0 Ó ¡! two rays P0Pi and P0Pj is also i; j (i; j = 1; 2; ¢ ¢ ¢ ; n). Assume that ®i (i = 1; 2; ¢ ¢ ¢ ; n) denote ¡¡¡!0 0 the unit vector of the vector P0Pi (i; j = 1; 2; ¢ ¢ ¢ ; n), then the unit vectors ®1; ®2; ¢ ¢ ¢ ; ®n No. 2 The law of Sines for an n-simplex in hyperbolic space and spherical space ¢ ¢ ¢ 217 are also independence. So the Gram matrix Q0 of the unit vectors ®1; ®2; ¢ ¢ ¢ ; ®n is positive and it is easy to know that 0 < det Q0 6 1. Therefore, this de¯nition is sensible. Remark Especially two-dimensional space angle of two-dimensional hyperbolic sim- plex (that is hyperbolic triangle) is just interior angle of hyperbolic triangle. So the n- dimensional space angle of n-dimensional simplex in n-dimensional hyperbolic space Hn(K) is extension of interior angle of hyperbolic triangle. De¯nition 2.2 (see [12]) Suppose that §n(H) = fP0;P1; ¢ ¢ ¢ ;Png be an n-dimensional simplex in n-dimensional hyperbolic space Hn(K), its volume V is the real number satisfying p (¡1)n sinh2 ¡KV = det(¤ (H)); (2.2) 2n ¢ (n!)2 n p n where ¤n(H) = (cosh ¡Kaij)i;j=0. Theorem 2.1 Suppose that §n(H) = fP0;P1; ¢ ¢ ¢ ;Png be an n-dimensional simplex in the n-dimensional hyperbolic space Hn(K), we have Q p Q p sinh ¡Kaij sinh ¡Kaij 16i<j6n 06i<j6n;i;j6=1 = = ¢ ¢ ¢ sin P sin P Q 0 p Q p1 sinh ¡Kaij sinh ¡Kaij 06i<j6n¡1 06i<j6n = = n p ; (2.3) sin Pn 2 2 ¢ n! ¢ sinh ¡KV where de¯nitions of sin Pi (i = 0; 1; ¢ ¢ ¢ ) are the same as De¯nition 2:1.