On the characteristic of projectively invariant Pseudo-distance on Finsler spaces

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1 IJST (015) 39A: Ianian Jounal of Siene & Tehnology On the haateisti of pojetively invaiant Pseudo-distane on insle spaes B. Bidabad* and M. Sepasi aulty of Mathematis and Compute Siene, Amikabi Univesity of Tehnology (Tehan Polytehni), Hafez Ave., Tehan, Ian s: & Abstat A pojetive paamete of a geodesi as solution of etain ODE is defined to be a paamete whih is invaiant unde pojetive hange of meti. Using pojetive paamete and Poinaé meti, an intinsi pojetively invaiant pseudo-distane an be onstuted. In the pesent wok, solutions of the above ODE ae haateized with espet to the sign of paallel Rii tenso on a insle spae. Moeove, the Rii tenso is used to define a insle stutue and it is shown that, the pseudo-distane is tivial on omplete insle spaes of positive semidefinite Rii tenso and it is a distane on a insle spae of paallel negative definite Rii tenso. Keywo: insle meti; Shwazian deivative; Rii tenso; pojetive paamete; pseudo-distane 1. Intodution If any geodesi of two insle spaes (M, )and (M, ) oinide as a set of points, then and d ae said to be pojetively elated. It is well-known that two insle spaes ae pojetively elated, if and only if thee is a 1-homogeneous sala field P(x, y) alled the pojetive fato satisfying G i(x, y) = G i (x, y) + P(x, y)y i, whee G i and G iae the oesponding spay veto fiel. In geneal the paamete "t" of a geodesi γ x i (t) on (M, ) does not emain invaiant unde pojetive hange of metis. A paamete whih emains invaiant unde pojetive hange of metis is alled pojetive paamete. The pojetive paamete is defined fist fo geodesis of geneal affine onnetion, (Thomas, 195; Eisenhat, 197; Bewald, 1937). In (Bidabad & Sepasi, 015), this paamete is aefully spelled out fo geodesis of insle spaes by the pesent authos as solutions of the following diffeential equation. {p, s} d 3 p 3 dp 3 dp [ ] dp = Ri n 1 jk (x(s), dx ) dxj dx k, (1) *Coesponding autho Reeived: Otobe 013 / Aepted: 31 Mah 015 whee, {p, s} is known in the liteatue as Shwazian deivative and "s" is the a-length paamete of γ. The Shwazian deivative is defined to be an opeato whih is invaiant unde all linea fational tansfomations t at+b f+d whee, ad b 0. That is { af+b, t} = {f, t}. () f+d By means of the equation (), the pojetive paamete is unique up to all linea fational tansfomations. In (Bidabad & Sepasi, 015), the pojetive paamete and the Poinaé meti ae used to define a pojetively invaiant pseudodistane denoted by d M and it is shown that in a omplete Einstein-insle spae with negative onstant Rii sala, the pojetively invaiant pseudo-distane is a onstant multiple of the insleian distane. Reall that two insle stutues and ae said to be homotheti if thee is a onstant λ suh that = λ ; as a oollay, the following esults ae obtained; Theoem A. Let (M, ) and (M, ) be two omplete Einstein insle spaes with Ri ij = g ij, and Ri ij = g ij, espetively, if and ae pojetively elated then they ae homotheti. Theoem B. Let (M, )and (M, ) be two omplete insle spaes of onstant negative flag uvatue, if

2 IJST (015) 39A: and ae pojetively elated then they ae homotheti. The last esult is also obtained by Shen (001) using anothe tehnique of poof. In the pesent wok, solutions of the diffeential equation (1) ae haateized with espet to the paallel Rii tenso in any of the Bewald, Chen o Catan onnetions as follows. Theoem 1. Let (M, ) be a insle spae of paallel Rii tenso. Then the Rii tenso is onstant along geodesis, and solutions of (1) ae lassified as follows. i) If {p, s} =, with > 0, then p = α os(s)+ β sin(s) γ os(s)+δ sin(s). (3) ii) If {p, s} = with > 0, then p = αes +βe s γe s +δe s. (4) iii) If {p, s} = 0, then p = α+βs γ+δs. (5) Next a new appoah to the study of pseudodistanes is established and the following esults ae obtained. Theoem. Let (M, ) be a onneted omplete insle spae of positive semi-definite Rii tenso. Then the intinsi pojetively invaiant pseudodistane is tivial, that is, d M = 0. Theoem 3. Let (M, ) be a onneted (omplete) insle spae of negative-definite paallel Rii tenso with espet to the Bewald o Chen onnetion. Then the intinsi pojetively invaiant pseudo-distaned M is a (omplete) distane. These esults ae genealizations of Riemannian woks (Kobayashi, 1978) and (Kobayashi & Sasaki, 1978) and establish a new appoah to the study of pojetive geomety in insle spaes.. Peliminaies A (globally defined) insle stutue on a diffeential manifold M is a funtion on the tangent bundle : TM [0, ) with the following popeties, i) Regulaity: is C on the entie slit tangent bundle TM 0, ii) Positive homogeneity: (x, λy)=λ(x, y) fo all λ > 0, iii) Stong onvexity: The Hessian matix (g ij ) = ([1/ ] y i y j) is positive-definite ontm 0 = TM\0. The pai (M, ) is known as a insle spae. Evey insle stutue indues a spay G = y i x i Gi (x, y) yi on TM, whee G i (x, y): = 1 gil {[ ] x k y lyk [ ] x l}. G is a globally defined veto field on TM. The pojetion of a flow line of G on M is alled a geodesi. The diffeential equation of a geodesi in the loal oodinates is given by d x i + G i (x(s), dx ) = 0, whee the paamete(t) = t (γ, dγ )d is a length paamete. Eveywhee t 0 d in this wok, the diffeential manifold M is supposed to be onneted. o a non-null, y T x M, the Riemann uvatue x i, G i y k y j Ry: T x M T x M is defined by R y (u) = R k i u k whee 1 G i R k i (y): = Gi x k 1 G i y k x j yj + G j y j G j y k. The Rii sala is defined by Ri = R i i, see fo instane (Akba-Zadeh, 1988) o (Bao, Chen, & Shen, 000). In the pesent wok, we use the definition of Rii tenso intodued by Akba- Zadeh as, Ri ik = 1/( Ri) y i yk. Moeove, by homogeneity we have Ri ik l i l k = Ri. Let us onside the spayg i = γ i jk y j y k, whee i γ jk 1 gis ( g sj x k g jk N i j 1 N k i y G i y j, li yi x s + g ks x j ), and, k), (Bao, Chen and Shen 000). l δ δx i = li ( x i 3. Pojetive paamete on Rii paallel insle spaes Let the Rii tenso of (M, ) be paallel with espet to any of the Catan, Bewald o Chen onnetions. We eall the Abel's identity in odinay diffeential equations. Conside the seond-ode linea odinay diffeential equation: y + P(x)y + Q(x)y = 0. (6) Conside the two linealy independent solutions, y 1 (x) and y (x). Then, the Wonskian of y 1 and y, w(y 1, y ): = y 1 y y 1 y satisfies w + Pw = 0, theefoe, w = w 0 e P(x)dx. (5) Poposition 1. If y 1 and y ae linealy independent solutions of the odinay diffeential equation y + Q(s)y(s) = 0, (8)

3 35 IJST (015) 39A: whee Q(s) = Ri n 1 jk (x(s), dx ) dxj dx k, then the geneal solution of (1) is given by u(t) = αy 1+ βy γy 1 + δy, (9) whee, αδ βγ 0. Poof: Aoding to (), it suffies to show that y 1 /y is a solution of (1). The tem P(x) in (7) is zeo, hene the Wonskian w(y 1, y ) is onstant. We may assume that w(y 1, y ) = 1. Then, u = 1/y, u u = y y and ( u ) = y y + (y ) u y = y ( y ), thus y y u u (u ) (u ) = ( Q(s))y (s) + 1 (u ), y (s) u u ( u ) = Q(s) + 1 (u ), u u u u 3 (u ) = Q(s). u u This ompletes poof of the poposition. Poof of Thoeem 1. Assume that the Rii tenso is paallel with espet to the Catan onnetion. Let us denote the hoizontal and vetial Catan ovaiant deivatives by δ δx k and espetively. Hene y k δ Ri ij = δri ij Ri δx k δx iγ k jk Ri j Γ ik = 0, (10) Ri ij = Ri ij Ri y k y k i A jk Ri A ik j = 0, (11) i whee, Γ jk = 1 gis ( δg sj δg jk + δg ks ) and Ai δx k δx s δx j jk g ih A hjk = g ih g hj 4 yk ae the oeffiients of Catan tenso. Conside the geodesi, γ = x i (s), whee"s" is the a-length paamete. Contating (10) by dxi gives dx i dx k ( Ri ij x k Nl k dx i dx k (Ri iγ jk ) dxi Ri ij ) y l dx k By means of (11) and, y j i A jk dx k (Ri iγ jk ) (Ri j Γ ik ) = 0. = 0, we have dx i dri ij Theefoe dri ij dx i and we have Ri ij dx i dxi dx k A il + Ri j ) dxi Ri ij d x i A jl Nl k (Ri i dx k Ri jγ ik = Ri ij d x i = 0. = onstant. (1) ollowing the method just used, we an pove that if the Rii tenso is paallel with espet to the Bewald o Chen onnetion, then along the geodesi γ paameteized by a-length, we have dx Ri i = onstant. ij Consideing the above assetion and Lemma 1, the equation (1) edues to a seond ode ODE with onstant oeffiients. Theefoe, sign of Rii tenso expliitly detemines a pojetive paamete "p" as an elementay funtion of "s" given by (3), (4)and (5). This ompletes the poof of Theoem. 4. Positive semi-definite Rii tenso Let I = ( 1, 1) be an open inteval with the Poinaé meti I = 4du (1 u ). The Poinaé distane between two points a and b in I is given by ρ(a, b) = ln (1 a)(1+b), (13) (1 b)(1+a) (Okada, 1983). A geodesi f I M on the insle spae (M, ) is said to be a pojetive map, if the natual paamete u on I is a pojetive paamete. We now ome to the main step fo defining the pseudo-distane d M on (M, ). We poeed in analogy with the teatment of Kobayashi in Riemannian geomety, (Kobayashi, 1978). Although he has onfimed that the onstution of intinsi pseudo-distane is valid fo any manifold with an affine onnetion, o moe geneally a pojetive onnetion (Kobayashi, 1977), we estit ou onsideation to the pseudo-distanes indued by the insle stutue on a onneted manifold M. Given any two points x and yin (M, ), we onside a hain αof geodesi segments joining these points. That is a hain of points x = x 0, x 1,, x k = y on M; pais of points a 1, b 1,, a k, b k in I;

4 IJST (015) 39A: pojetive maps f 1,, f k, f i I M, suh that f i (a i ) = x i 1, f i (b i ) = x i, whee, i = 1,, k. By vitue of the Poinaé distane ρ(.,. )on I we define the length L(α) of the hain α by L(α) = i ρ(a i, b i ), and we put d M (x, y): = infl(α), (14) whee the infimum is taken ove all hains αof geodesi segments fom x to y. Poposition. Let (M, ) be a insle spae. Then fo any points x, y, and z in M, d M satisfies i. d M (x, y) = d M (y, x). ii. d M (x, z) d M (x, y) + d M (y, z). iii. If x = y then d M (x, y) = 0, but the invese is not always tue. Taditionally, d M (x, y) is alled the pseudodistane of any two points x and y on M. om the popety () of Shwazian deivative, and the fat that the pojetive paamete is invaiant unde fational tansfomation, the pseudo-distane d M is pojetively invaiant. Poof of Theoem. In ode to pove Theoem we need the following Lemmas. Lemma 3. Let (M, ) be a omplete insle spae. Conside two points x 0 and x 1 on M. If thee exists a geodesi x(u) with pojetive paamete u, 1 < u < +1, suh thatx 0 = x(u 0 ) and x 1 = x(u 1 ) fo some u 0 and u 1 in R then d M (x 0, x 1 ) = 0. Poof: Linea equation of the segment passing though the points (u 0, 1/)and (u 1, 1/) is given by û = u 1 (u 1 +u 0 ). Hee, û is a linea u 1 u 0 u 1 u 0 tansfomation of u and is also a pojetive paamete. We have 1 < û < 1 wheneve u 0 < u < u 1. Next, we onside the hain α of pojetive maps, a n and b n whee f n = x(nû), a n = 1 n, b n = 1 n. Note that f n ( 1 n ) = x (n ( 1 n)) = x ( 1 ) = x(u 0 ) and ρ ( 1, 1 n n ) = ln (1+ 1 n )(1+ 1 n ) (1 1 n )(1 1 ). n Consideing n suffiiently lage, we have d M (x 0, x 1 ) = infl(α) = 0. This ompletes the poof of Lemma 3. Lemma 4. Let (M, ) be a omplete insle spae and x(s) a geodesi with a-length paamete < s <. Assume that thee exists a (finite o infinite) sequene of open intevals I i = (a i, b i ), i = 0, ±1, ±, suh that; i) a i+1 b i, lim i a i =, lim i + b i = + and i I i = (, + ) ; ii) in eah inteval I i = (a i, b i ), a pojetive paamete "u" moves fom to + wheneve t moves fom a i to b i. Then, fo any pai of points x 0 and x 1 on this geodesi, we have d M (x 0, x 1 ) = 0. Poof: By means of Lemma1, the distane between any two points in the same inteval I i is zeo. Two onseutive open intevals I i and I i+1 have eithe a point as a bounday point o an inteval in ommon. In eah ase, given ϵ > 0, thee exist the points S i and S i+1 in I i and I i+1 espetively suh that d M (x(s i ), x(s i+1 )) < ϵ. This ompletes the poof of Lemma 4. The following Lemmas pemit us to onstut the open intevals I i in Lemma 3. The poofs ae given in (Kobayashi & Sasaki, 1978). Lemma 5. In the ODE (8), if Q(s) = 0 fo all s R then evey solution y(s) has at least one zeo unless Q(s) = 0 and y(s) is onstant 0. In the sequel the Stum's sepaation theoem whih laims; given a homogeneous seond ode linea diffeential equation and two ontinuous linea independent is needed. Solutions v(x) and u(x) with x 0 and x 1 suessive oots of v(x), so u(x) has exatly one oot in the open inteval (x 0, x 1 ). Lemma 6. Let y 1 (s)and y (s) be two linealy independent solutions of (8). If a and b ae two onseutive zeos of y then u = y 1 (s)/y (s) o u = y 1 (s)/y (s) is a pojetive paamete on the inteval (a, b) whih moves fom to + as s moves fom a to b. The diffeential equation (8) is said to be osillatoy at s = ± if the zeos < a < a 1 < a 0 < a 1 < a <, of the solution y(s) have the popety lim h a h = and lim h + b h = +. Then the sequene of intevals I i = (a i, b i ) satisfies the ondition of Lemma 3. This fat poves Theoem in this ase. Next, we onside the ase that (8) is nonosillatoy at s = +. That is, y (s) does not vanish fo suffiiently lage s. Aoding to the Stum's theoem, this ondition is independent of hoie of a patiula solution y (s).

5 37 IJST (015) 39A: Lemma 7. If the diffeential equation (8) is nonosillatoy at s = +, then thee is a solution y (s) whih is uniquely detemined up to a onstant fato satisfying y lim (s) s + = 0, (15) y 1 (s) fo any solution y 1 (s) linealy independent of y (s). The solution y (s) in Lemma 6 is alled a pinipal solution. Hee, we onside a weake vesion of ompaison Theoem of Stum as follows. Lemma 8. Conside two diffeential equations (i) y (s) + Q 1 (s)y(s) = 0, (ii) y (s) + Q (s)y(s) = 0, with Q 1 (s) > Q (s). Let y 1 (s) and y (s) be solutions of (i) and (ii) espetively suh that y 1(a) y (a). (16) y 1 (a) y (a) If y 1 (s) and y (s) have no zeo in the inteval a < s < +, then fo s > a y 1(s) y (s). (17) y 1 (s) y (s) If y (a) = 0, then the tem y (a) is onsideed to y (a) be. One an efe to (Du and Kwnog, 1990) and (Kobayashi and Sasaki, 1978) fo moe details about this subjet. Lemma 9. Assume that the diffeential equation (8) is nonosillatoy at s = + and that Q(s) 0. Let y(s) be a pinipal solution as in Lemma 6. If a is the lagest zeo of y (s) and if y 1 (s) is a solution linealy independent of y (s), then y 1 (s) vanishes at some s > a. We ae now in a position to omplete the poof of the theoem whee the diffeential equation (8) is nonosillatoy at s = +, o s =. If (8) is non-osillatoy at s = + but osillatoy at s =, we take a pinipal solution y (s) and anothe solution y 1 (s) linealy independent of y (s). Let <a < a 1 < a 0 < a 1 < a <, be the zeos of y (s). Then the sequene of intevals, I i = (a i, b i ), fo i =,, 1,0,1,,, k with a k+1 = +, equipped with a pojetive paamete u = y 1 o u = y 1 satisfy the equiements of y y Lemma 3. We note that Lemma 8 implies that u is a pojetive paamete in the last intevali k = (a k, + ). If (8) is nonosillatoy at s = but osillatoy at s = +, we eplae Lemma 6 and Lemma 8 by the analogous Lemmas fo s =. Assume that (8) is nonosillatoy at ±. Let y (s) be a pinipal solution fo s = + and not fo s =. Let y 1 (s) be a pinipal solution fo s =. then y 1 (s) and y (s) ae linealy independent. We obtain a sequene of intevals I i, i = 0, 1,, k with a pojetive paamete u = y 1 y, u = y 1 y, u = y y 1 o u = y y 1 satisfying the equiements of Lemma 3. In this ase, thee ae some ovelaps among these intevals. If y (s) is a pinipal solution fo both s = + and s = then we onside y 1 (s) as a solution linealy independent of y (s). We obtain a sequene of intevals I i, i = 0, 1,, k, with a pojetive paamete u = o u = y 1 satisfying y the equiements of Lemma 3. In this ase, thee ae no ovelaps of intevals. This ompletes the poof of Theoem. 5. Paallel negative-definite Rii tenso We eall the following theoem whih will be used in the sequel. Theoem A. Let (M, ) be a onneted (omplete) insle spae fo whih the Rii tenso satisfies, Ri ij g ij, as maties, fo a positive onstant. Then d M is a (omplete) distane. (Bidabad & Sepasi, 015). Poof of Theoem 3. Let us onside the insle stutue defined by means of the Rii tenso as (x, y) = Ri ij (x, y)y i y j. One an easily hek that satisfies all popeties of a insle stutue on M. Moe peisely; i) by definition is C on the entie slit tangent bundle TM 0 ; ii) The Rii tenso Ri ij (x, y) is 0-homogeneous, hene (x, λy) = λ (x, y) fo all λ > 0; iii) again, aoding to the 0-homogeneity of Ri ij (x, y), though staightfowad alulation we get, ĝ ij = [ 1 ] y i y j = Ri ij(x, y). The Rii tenso is supposed to be negativedefinite thus the Hessian matix (ĝ ij ) is positivedefinite. Next, we show that the spay oeffiients of and ae equal, that is, Ĝ i = G i, hene we have

6 IJST (015) 39A: Ĝ i = 1 ( Ri)ih ( y h x j yj x h) = 1 ( Ri)ih ( ( Ri l y l y ) y h x j y j ( Ri ly l y ) x h ) = 1 ( Ri)ih ( (Ri hly l ) x j y j + (Ri l ) x h y l y ) = Ri ih (Ri hly l ) x j y j 1 Riih (Ri l ) x h y l y. (18) Let Rii tenso be paallel with espet to the Bewald onnetion b. Simila aguments hold well fo Chen onnetion. We have b δ Ri hl = δri hl Ri δx j δx j h G lj Ri l G hj = 0, G lj = 1 G b y k y l y j. (19) Ri ij = Ri ij yk = 0. (0) Contating (19) with Ri ih y j y l lea to Ri ih y j y l Ri hl x j Ri ih y j y l Ri hl x j On the othe hand Ri ih Ri ha G a 1 G a Riih Ri la y h yl = 0. G i 1 Riih G Ri a la y h yl = 0. (1) insle geomety, Spinge-Velag. Bewald, L. (1937). On the pojetive geomety of paths, Po. Edinbugh Math. So.5, Bidabad, B., & Sepasi, M. (015). On a pojetively invaiant pseudo-distane in insle geomety, Intenational Jounal of Geometi Metho in Moden Physis. 1(4), (1 pages). Du, M. & Kwnog, M. (1990). Stum ompaison theoems fo seond-ode delay equations, Jounal of Mathematial Analysis and Appliation, 15, Eisenhat, L. P. (197). Non-Riemannian geomety, Ameian Mathematial Soiety Colloquium Publiations, 8. Kobayashi, S. (1977). Intinsi distanes assoiated with an affine o pojetive stutue, J. a. Si. Univ. of Tokyo, IA, 4, Kobayashi, S. (1978). Pojetive stutue of hypeboli type, Minimal Sub-manifol and geodesis, Kaigai Publiations, Tokyo, Kobayashi, S., & Sasaki, T. (1978). Pojetive stutues with tivial intinsi pseudo-distane, Minimal submanifol and geodesis, Kaigai Publiations, Tokyo, Okada, T. (1983). On models of pojetively at insle Spaes of onstant negative uvatue, Tenso, N. S. 40, Sepasi, M., & Bidabad, B. (014). On a pojetively invaiant distane on insle spaes of onstant negative Rii sala, C. R. Aad. Si. Pais, Se. 135(014) Shen, Z. (001). On pojetively elated Einstein metis in Riemann-insle geomety, Math, Ann. 30, Thomas, T. Y. (195). On the pojetive and equipojetive geometies of paths, Po. N. A. S Riih y y l Ri l x h 1 Riih y y l Ri l x h + 1 Riih y y l a Ri la G h + 1 Riih y y l a Ri a G lh = Riih y G Ri a a yh = 0. () Consideing, (18), (1) and () we have Ĝ i = G i. As a onsequene, we have Ri ij = Ri ij. On the othe hand, we just asseted that ĝ ij (x, y) = Ri ij (x, y). Thus, we have ĝ ij (x, y) = Ri ij (x, y). Aoding to TheoemA, d Mis a (omplete) distane. The two spaes (M, ) and (M, ) ae affine and we have d M = d M. Hene, d M is a (omplete) distane. Refeenes Akba-Zadeh, H. (1988). Su les espaes de insle _a oubues setionnelles onstant, Aad. Roy. Belg. Bull. CI. Si., 5(14), Bao, D., Chen, S. S., & Shen, Z. (000). Riemann-