Fundamental theorem of hyperbolic geometry without the injectivity assumption

F undamen tal theorem of h yp erb olic geometry withou t the injectivity assumption GUO WU Y A O Departmen t of Mathematical Sciences , Tsingh ua Univ ersit y Beijing, 100084, P .R. China e-mail: gwyao@math. tsinghua.edu.cn A BSTRA CT. Let H n be the n − dimensio nal hyperb olic spa ce. It is well known that, if f : H n → H n is a bijection tha t pr eserves r − dimensio nal hyperplanes, then f is an isometr y . In this pa p er we make neither inj ectivity nor r − hyper plane preserving a ssumptions on f and prov e the follo wing result: Suppo se that f : H n → H n is a surjective map and maps an r − hyperplane in to an r − hyper plane, then f is an isometry . The E uclidean v ersion was obtained by A. Chubarev a nd I. Pinelis in 199 9 amo ng other things. Our pro of is ess e ntially different from their and the similar problem arising in the s pher ical case is op en. 1 . In tro duction Let ˆ R n = R n ∪ {∞} wh er e R n is th e n − d imensional Euclidean space and let H n b e the n − dimensional h yp erb olic space. A map f of ˆ R n to itself is called r − sphere p reserving if f maps an r − dim en sional sphere on to an r − dimensional sphere. Similarly , a map f of R n (or H n ) to R n (or H n ) is called r − h yp erplane preserving if f maps an r − di- mensional hyp erplane onto an r − dimensional h yp erplane in R n (or H n ). In p articular, when r = 1, w e call the corresp ondin g map f to b e a circle-preservin g (line-preserving, geod esic-preserving) map in ˆ R n ( R n , H n ), r esp ectiv ely . I n th e s equel, we pr escrib e n ≥ 2 and 1 ≤ r < n . The prop ert y of a M¨ obius transf orm ation acting on ˆ C is so clear and the relations b et we en M¨ obius transformation and s ome of its prop er ty hav e b een extensivel y studied. F or examples, Carath ´ eo dory fir st pro v ed th at if f : ˆ C → ˆ C is a circle-preserving bijection, then f is a M¨ obius transformation (see [4 ] or [12]); Nehari [11] sho w ed that if f : ˆ C → ˆ C is a non-constant meromorphic function that preserve s circles, then f is a M¨ obius transformation. Of course, the analogous p roblem for affine (or isometric) transformations on R n (or H n ) is also concerned. In [7], Jeffers obtained the f ollo w in g extension of C arath ´ eo dory ’s 2000 Mathematics Subje ct Classific ation. Primary 37B05, 30C35; Secondary 51F15. Key wor ds and phr ases. M¨ obius transformation, affine transformation, isometric transformation. The author wa s supp orted by a F ound ation for th e Auth or of N ational Excellent Do ctoral Dissertation (Grant No. 20051 8) and the N ational Natural Science F ound ation of China. 1 2 GUO WU Y A O result to all three cases (for concision, w e com b ine three theorems obtained b y him in to one). Theorem A. Supp ose that f : ˆ R n → ˆ R n ( R n → R n , H n → H n ) is a bije ction that pr eserves r − dimensional spher es ( r − dimensional hyp erplanes). Then f is a(n) M ¨ obius (affine, i sometric) tr ansformation. An r − sphere p reserving map f is called degenerate if its image f ( ˆ R n ) is an r − di- mensional sph ere; otherwise, f is called non-d egenerate. The reader will easily guess the prop er defin itions for non-d egenerate and degenerate m aps in the Euclidean and hyper b olic settings. In a recent article [8], B. Li and Y. W ang made neither injectivit y n or su rjectivit y assumptions on f and pro v ed Theorem B. Supp ose that f : ˆ R n → ˆ R n ( R n → R n , H n → H n ) is a cir c le- pr eserving (line-pr eserving, ge o desic-pr eserving) map. Then f is a(n) M¨ obius (affine, isometric) tr ansformation if and only if f is non-de ge ner ate. The existence of degenerate maps was sho wn in [8, 14]. More recen tly , the auth or join t with B. Li [9] obtained the follo wing generalization of Theorem B. Theorem C. Supp ose that f : ˆ R n → ˆ R n ( R n → R n , H n → H n ) is an r − spher e pr e se rvi ng ( r − hyp erplane pr eserving) map. Then f is a(n) M ¨ obius (affine, isometric) tr ansformation if and only if f is non-de g ener ate. In [5], C h ubarev and Pinelis sho w ed, among other things, that the injectiv e condition for the Euclidean case R n in Th eorem A can b e remo v ed. Precisely , the follo wing theorem w as im p lied. Theorem D. Supp ose that f : R n → R n is a surje ctive map and maps every r − dimen- sional hyp erplane into an r − dimensional hyp erplane. Then f is an affine tr ansformation. Inspired b y Theorem D, the follo wing tw o conjectures w ere naturally p osed in [9]: Conjecture 1. Supp ose that f : ˆ R n → ˆ R n is a surje ctive map and maps ev e ry r − dimen- sional spher e i nto an r − dimensional spher e. Then f is a M¨ obius tr ansformation. and Conjecture 2. Supp ose that f : H n → H n is a surje ctive map and maps e very r − dimen- sional hyp erplane into an r − dimensional hyp e rplane. Then f is an isometric tr ansforma- tion. The aim of this pap er is to pro v e Conj ecture 2 by app lying Theorem C but lea ve Conjecture 1 op en . F or completeness, we also giv e a simple pro of of Theorem D in S ection 5. Other results in the line can b e foun d in [1, 2, 5, 6, 10, 13]. R emark 1 . Recen tly , the author [15] pro v ed that Conjecture 1 is tru e in the case r = n − 1. F undamenta l th eorem of hyp erb olic geometry 3 2 . Some preparations This section is d ev oted to reduce the p ro of of Conjecture 2 to th at of the sp ecial case when r = 1. That is, w e only need to pr o v e that, Theorem 1. Supp ose that f : H n → H n is a surje ctive map and maps every ge o desic into a ge o desic. Then f is an isometric tr ansformation. This r eduction clearly dep end s on the follo w ing lemma. Lemma 1. Supp ose (i) ther e exists some r such that the map f : H n → H n maps every r − dimensional hyp erplane into an r − dimensional hyp erplane, (ii ) f ( H n ) is not c ontaine d in an r − dimensional hyp erplane. Then f or any given k -dimensional hyp erplane Γ ⊆ H n ( 1 ≤ k ≤ r ), f maps Γ into a k - dimensional hyp erplane. In p articular, f maps a ge o desic into a ge o desic. Throughout our discu s sion, lo w er case letters will denote p oin ts, upp er case letters s ets of p oin ts, subscripts for lik e ob jects, and primes for images und er the map f . The n otable exception to these conv entions will b e w h en the image f (Λ) of a set Λ is not p resumed to b e Λ ′ but w e will write f (Λ) ⊆ Λ ′ . F or a n onempt y sub set A with # A ≥ 2 in H n , let Q A d en ote the t − d imensional h yp erplane con taining A such that t is the smallest p ositiv e integ er. It is easy to see that Q A and t are un iqu ely determined by the set A . No w, w e pro v e Lemma 1: If r = 1, it is a fortiori. Let r ≥ 2 and k = r − 1. Em b edd in g S into some r − d imensional h yp erplane Γ, we ha v e Γ ′ = f (Γ) as an r -dimensional hyp erplane by h yp othesis. S ince f ( H n ) is not con tained in an r − dimensional hyperp lane, we can find a p oin t p ∈ H n \ Γ suc h that p ′ = f ( p ) 6∈ Γ ′ . Letting Γ 1 = Q { S, p } , then Γ 1 is an r − dimen sional hyp erplane. Set Γ ′ 1 = f (Γ 1 ). Since f ( S ) = f (Γ ∩ Γ 1 ) ⊆ f (Γ) ∩ f (Γ 1 ) and f (Γ) ∩ f (Γ 1 ) = Γ ′ ∩ Γ ′ 1 is contai ned in an ( r − 1)-dimensional hyp erplane, the lemma holds for k = r − 1. It is clear that f ( H n ) is also n ot con tained in an ( r − 1) − dimensional h yp erplane. Thus, we can in ductiv ely bac kw ard prov e that if this lemma holds for k ( ≥ 2), then it do es for k − 1. This lemma then follo ws. 3 . Pro of of Th eorem 1 In this section, w e prov e Th eorem 1. Throughout this section except in Lemma 2, w e assume that f sat isfies th e conditions of Theorem 1. l xy alw a ys den otes th e geodesic determined b y x an d y in H n . Lemma 2. Supp ose f : H n → H n maps a ge o desic into a ge o desic. Then f maps an r − dimensional hyp e rplane into an r − dimensional hyp erplane for 1 ≤ r ≤ n − 1 . Pr o of. W e use induction. Let n ≥ 3 and assume that f m aps an r − d imensional hyp er p lane in to an r − dimensional h yp erplane for some r ∈ [1 , n − 2]. W e need to show that f maps an ( r + 1) − dimensional h yp erplane int o an ( r + 1) − dimensional hyp erplane. 4 GUO WU Y A O Supp ose not. Then there exists an ( r + 1) − dimensional hyp erplane S s u c h that Q f ( S ) has dimension d ≥ r + 2. Ther efore, there exist r + 3 p oin ts { p ′ 1 , p ′ 2 , · · · , p ′ r +3 } in f ( S ) such that no ( r + 1) − dimensional hyperp lane contai ns them and the h yp er p lane Q K ′ spanned b y K ′ = { p ′ 1 , p ′ 2 , · · · , p ′ r +3 } has dimension r + 2. On the other hand, there exist r + 3 distinct p oin ts { p 1 , p 2 , · · · , p r +3 } in S su c h that f ( p i ) = p ′ i ( i = 1 , 2 , · · · , r + 3). It is clear that no r − dimensional hyp er p lane conta ins more than r + 2 p oint s of { p 1 , p 2 , - · · · , p r +3 } by the inductiv e assumption. Therefore, ev ery r + 1 p oin ts of { p 1 , p 2 , · · · , p r +2 } can span a u nique r -dimensional h yp erp lane and these r + 2 spann ed hyper p lanes divide the ( r + 1) − d imensional hyp erplane S into 2 r +2 − 1 disjoint parts. The p oin t p r +3 is lo cated inside some p art. Observe that { p 1 , p 2 , · · · , p r +2 } frames an ( r + 1)-simplex in S . An yw a y there exists at least a p oin t of { p 1 , p 2 , · · · , p r +2 } , sa y p 1 , suc h th at the geo desic l p 1 p r +3 crosses the r − dimens ional hyperp lane Λ = Q { p 2 , p 3 , · · · , p r +2 } . Letting q = l p 1 p r +3 ∩ Λ, then q 6 = p 1 and q ′ = f ( q ) ∈ Q { p ′ 2 , p ′ 3 , · · · , p ′ r +2 } b y the inductive assump tion. Th us, p ′ r +3 = f ( p r +3 ) ∈ f ( l q p 1 ) which sho ws that p ′ r +3 ∈ Q { p ′ 1 , p ′ 2 , · · · , p ′ r +2 } since f ( l q p 1 ) ⊆ l q ′ p ′ 1 . This further ind icates that { p ′ 1 , p ′ 2 , · · · , p ′ r +3 } is con tained in the ( r + 1) − dimensional hyp erplane Q { p ′ 1 , p ′ 2 , · · · , p ′ r +2 } , a con tradiction. Th e in ductiv e pro of is completed. R emark 2 . Lemm a 2 can b e regarded as a con ve rse of Lemma 1. W e hav e an essen tial difficult y in obtaining its sph erical v ersion wh ich is also the only bug to solv e Conjecture 1 while we pro v e Conjecture 1 when r = n − 1 in [15]. In other w ords, the answ er to the follo w ing problem is crucial to the solution of C onjecture 1. Problem 1. Supp ose tha t f : ˆ R n → ˆ R n ( n ≥ 3 ) is a surje ctive map and maps a ci r cle into a cir cle. Can we say that f maps an ( n − 1) − dimensional spher e into an ( n − 1) − dimensional spher e ? Lemma 3. Supp ose D is a domain in H n . If f ( D ) is c ontaine d in an ( n − 1) − dimensional hyp erplane, then f is c onstant on D . Pr o of. Sup p ose not. Then f ( D ) is con tained in an ( n − 1) − dimensional hyperp lane, sa y Γ ′ ⊆ H n , and f ( D ) cont ains at least t w o p oints. Let S = { w ∈ H n : f ( w ) ∈ Γ ′ } . Ob viously , D ⊆ S , f ( S ) ⊆ Γ ′ and S 6 = H n . Claim 1. S is path-connected. W e may c ho ose t w o p oin ts p, q in S suc h that f ( p ) 6 = f ( q ) since f ( S ) = Γ ′ . No w, for an y other p oin t w ∈ S , it is no harm to assu me that f ( w ) 6 = f ( q ). Th us, the geo desic l w q ⊆ S since f ( l w q ) ⊆ Γ ′ whic h implies that S is path-connected. Claim 2. H n − S con tains n o in terior p oints. Supp ose to the con trary . Let p b e an in terior p oint of H n − S and P b e the largest connected op en set in H n \ S suc h that p ∈ P . Whence, ev ery set f ( l pq ∩ D ) is a sin gleton since otherwise p ′ ∈ Γ ′ , where the geo desic l pq passes through p and a p oin t q ∈ D . Since f ( D ) con tains at least tw o p oint s, there are t w o p oints, sa y u and v , suc h th at the geo desics l p ′ u ′ and l p ′ v ′ are d istinct. Therefore, there exists at least a p oint x in D and a sequen ce of p oin ts { x n } ∞ n =1 in D su c h that lim n →∞ x n = x and l p ′ x ′ n 6 = l p ′ x ′ , ∀ n. F undamenta l th eorem of hyp erb olic geometry 5 Observe that f ( l px n ∩ D ) = x ′ n and f ( l px ∩ D ) = x ′ and x ′ n 6 = x ′ for all n . W e then ma y c ho ose sufficien tly large int eger m and a p oint y m in l px m ∩ D suc h that the geo desic l xy m through x and y m crosses the domain P . Recalling that f ( x ) = x ′ and f ( y m ) = x ′ m , w e can fi nd a p oin t z ∈ l xy m ∩ P suc h that f ( z ) ∈ f ( l xy m ) ⊆ Γ ′ . This indicates th at z ∈ S , a contradict ion. W e con tin ue to derive a new con tradiction from the ab o v e tw o Claim s as follo ws. Let Λ ′ ⊆ H n \ Γ ′ b e an ( n − 1) − dimensional hyp erplane. Cho ose n p oin ts { p ′ 1 , p ′ 2 , · · · , p ′ n } in Λ ′ suc h that these n p oints are not conta ined in an ( n − 2) − dimens ional h yp erplane (when n = 2, suc h c hoice is trivial). There exist n distinct p oin ts { p 1 , p 2 , · · · , p n } in H n suc h that f ( p i ) = p ′ i ( i = 1 , 2 , · · · , n ). Let Λ = Q { p 1 , p 2 , · · · , p n } b e the hyp erplane spanned by { p 1 , p 2 , · · · , p n } . It is easy to deduce from Lemma 2 that the dimension dim (Λ) of Λ is just n − 1 and f (Λ) ⊆ Λ ′ . Notice that Λ divides H n in to t w o disjoin t domains. Necessarily , Λ ∩ S 6 = ∅ by Claim s 1 and 2. T h us, we hav e f (Λ ∩ S ) 6 = ∅ which contradicts that f (Λ) ∩ f ( S ) ⊆ Λ ′ ∩ Γ ′ = ∅ . Th is completes the p ro of of L emm a 3 . Lemma 4. Supp ose D is a do main in H n . Then f ( D ) c annot b e c ontaine d in an ( n − 1) − dimensional hyp e rplane in H n . Pr o of. Sup p ose not. Then by Lemma 3 , f is constan t on D , in other words, f maps D to a p oint, sa y p ′ ∈ H n . Let D b e the largest connected op en set of H n suc h that f ( D ) = { p ′ } . Set S = { w ∈ H n : f ( w ) = p ′ } . Obviously , D ⊆ D ⊆ S and S 6 = H n . Claim. H n − S con tains interior p oints. W e ma y c ho ose t w o ( n − 1) − d im en sional hyp erplanes Φ ′ and Ψ ′ in H n \{ p ′ } suc h that, (i) Φ ′ ∩ Ψ ′ = ∅ and (ii) th e conv ex set K ′ = { z ′ ∈ H n : ∃ x ′ ∈ Φ ′ , y ′ ∈ Ψ ′ , s.t. z ′ ∈ l x ′ y ′ } do es n ot conta in p ′ . By v ir tue of Lemma 2, one can find t wo hyp erplanes Φ and Ψ in H n suc h that f (Φ) ⊆ Φ ′ , f (Ψ) ⊆ Ψ ′ and dim (Φ) = dim (Ψ) = n − 1. I t is eviden t that Φ ∩ Ψ = ∅ . It is also clear th at the conv ex set K = { z ∈ H n : ∃ x ∈ Φ , y ∈ Ψ , s .t. z ∈ l xy } con tains in terior p oints and f ( K ) ⊆ K ′ . Th is claim follo ws immediately . No w, in virtue of th e ab o v e Claim , it is easy to c ho ose an in terior p oint q of H n \ S and a p oin t b on the b ound ary of D suc h that geo desics through p and p oin ts of D con tain a neigh b orho od N of b . Noticing that all suc h geod esics are mapp ed int o l q ′ p ′ , we h a v e N ⊆ S by Lemma 3. T his contradicti on establishes this lemma. Lemma 5. f maps every ( n − 1) − dimensional hyp erplane in H n onto an ( n − 1) − dimen- sional hyp erplane in H n , i.e. , f is ( n − 1) − hyp erplane pr eserving. Pr o of. Giv en an ( n − 1) − dimensional hyp erplane S in H n , by Lemma 2, th ere exists an ( n − 1) − dimensional hyper p lane S ′ ⊃ f ( S ). W e n ow s h o w that S ′ = f ( S ). S upp ose not, then there should exist some p oin t a ′ ∈ S ′ \ f ( S ). Let a ∈ H n \ S b e an inv erse image of a ′ under f . Th e collec tion of geo desics through a an d p oints of S cov ers a domain in H n whic h is mapp ed into the ( n − 1) − dimensional hyper p lane S ′ . It d eriv es a desired con tradiction from Lemma 4. Thus, we pr o v e that f is ( n − 1) − hyperp lane p reserving. Finally , the pr o of of Theorem 1 is concluded by Lemma 5 and Theorem C (let r = n − 1). 6 GUO WU Y A O 4 . A simple pro of of Theorem D By the foregoing reasoning, the pro of of Theorem D reduces to th at of th e follo win g theorem. Theorem 2. Supp ose that f : R n → R n is a surje ctive map and maps every line into a line. Then f is an affine tr ansforma tion. Pr o of. F or one thing, it is easy to prov e that f m ap s an ( n − 1)-dimensional hyperp lane in to an ( n − 1) − dimensional h yp er p lane as proving Lemma 2. W e claim that f also maps an ( n − 1)-dimensional h yp erp lane ont o an ( n − 1) − dimen- sional hyperp lane. S upp ose to the con trary . Then there exists an ( n − 1) − dimensional h yp erplane Γ in R n suc h th at f (Γ) is conta ined in an ( n − 1) − dimensional h yp erp lane Γ ′ and Γ ′ \ f (Γ) 6 = ∅ . Let p ′ ∈ Γ ′ \ f (Γ) and p an in v erse image of p ′ . Observe th at lines through p and p oin ts in R n \{ p } either cross Γ or p arallel Γ, and the formers are mapp ed in to Γ ′ and the latters are con tained in an ( n − 1)-dimensional h yp erp lane parallel to Γ and h en ce are mapp ed in to an ( n − 1)-dimensional hyp erplane, s a y Λ ′ . Thus, R n is m app ed into the union of Γ ′ and Λ ′ , a contradict ion. The pro of of Theorem 2 is completed by applyin g Th eorem C . 5 . Concluding remarks All results m en tioned in this p ap er b elong to a y oung and activ e geometrical discipline called “c haracterizations of geomet rical map p ings un d er mild h yp otheses”. The disci- pline started aroun d 1950 with fun damen tal theorems of A. D. Alexandro v on sp acetime transformations and causal automorphisms (see [3]). Through ou t th e conditions in these theorems, for examples, T h eorems A ∼ D and our main result, sur jectivit y , injectivit y and non-degenerate pla y in evitable roles. In our result, we remo v e the injectivit y assum ption but n on-degenerate one is satisfied automatically . In a futur e pap er [16], the author eve n further replaces the surjectivit y assum ption with the condition that ev ery r -dimensional h yp erplane conta ins at least r + 1 image p oin ts. On e ma y ask, wh at situation will b e if the su rjectivit y assump tion on f replaced by the injectivit y one? Actually , we can sa y nothing on f b ecause there exists a so-called degenerate map f : H n → H n suc h that f maps H n one-to-one into s ome ˜ r -d imensional hyperp lane. So, w e need another restriction on f to guarantee that it is an automorphism. Naturally , “non-degenerate” is the fi rst candidate. Ma yb e one exp ects a theorem as follo ws: If f : H n → H n is an inje ctive map and maps an r -dimensional hyp erplane into an r -dimensional hyp erplane and if f is non-de gener ate, i.e., f ( H n ) c annot b e c ontaine d in an r -dimensional hyp erplane, then f is an isometry. Unfortunately , recen tly in oral communicat ion, a coun terexample w as giv en by L i Baokui. W e interpret it h ere. Counter example: F or con v enience, let n = 2 and use the semisp here S in R 3 as the mo del of H 2 , namely , S = { ( x, y , z ) ∈ R 3 : x 2 + y 2 + z 2 = 1 , z > 0 } . F undamenta l th eorem of hyp erb olic geometry 7 Then, all geod esics in S are these semicircles p erp endicular to the X O Y -plane. Pr o ject S on to the unit disk D in R 2 : D = { ( x, y ) ∈ R 2 : x 2 + y 2 < 1 } . Sa y , let P denote the pro jection: P ( x, y , z ) = ( x, y ) . It is easy to see that th e images of geo desics in S under P are these segmen ts with ends at the b oundary of D . Let A b e an affine transf ormation in R 2 with the form : A ( x, y ) = ( ax, by ) , a, b ∈ (0 , 1) . Th us, the comp osition map f = P − 1 ◦ A ◦ P maps S into S . It is evid ent that f is injectiv e and non-degenerate. It maps a geo desic in S into a geod esic and the action of f on geo d esics is shortening them an d h ence f is not an isometry . Although the exceptional ph enomenon o ccurs in H n , w e cannot fi nd su ch coun terex- ample in ˆ R n or R n so far. Whence, we end remarks with t w o op en prob lems. Problem 2. Supp ose f : ˆ R n → ˆ R n is an inje ctiv e map and maps an r - dimensional spher e into an r -dimensional spher e. Can we say that f is a M ¨ obius tr ansformatio n if f ( ˆ R n ) c annot b e c ontaine d in an r - dimensional spher e? Problem 3. Supp ose f : R n → R n is an inje c tive map and maps an r -dimensional hy- p erplane into an r -dimensional hyp erplane. 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