The group of isometries of the French rail ways metric

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1 Stu. Univ. Babeş-Bolyai Math. 58(2013), No. 4, The group of isometries of the French rail ways metric Vasile Bulgărean To the memory of Professor Mircea-Eugen Craioveanu ( ) Abstract. In this paper we stuy the isometry groups Iso F,p (R n ), where F,p is the French rail ways metric given by (2.1). We erive some general properties of Iso F,p (R n ) an we iscuss the particular situation when X = R n an = 2 is the Eucliean metric. Mathematics Subject Classification (2010): 51B20, 51F99, 51K05, 51K99, 51N25. Keywors: Isometry with respect to a metric, group of isometries, French railroa metric, homeomorphism group of R n. 1. Introuction Let (X, ) be a metric space. The map f : X X is calle an isometry with respect to the metric (or a -isometry), if it is surjective an it preserves the istances. That is for any points x, y X the relation (f(x), f(y)) = (x, y) hols. From this relation it follows that f is injective, hence it is bijective. Denote by Iso (X) the set of all isometries of the metric space (X, ). It is clear that (Iso (X), ) is a subgroup of (S(X), ), where S(X) enotes the group of all bijective transformations f : X X. We will call (Iso (X), ) the group of isometries of the metric space (X, ). A general, important an complicate problem is to escribe the group (Iso (X), ). This problem was formulate in the paper [2] for metric spaces with a metric that is not given by a norm efine by an inner prouct. Some results in irection to solve this problem for some particular metrics are the following. D.J. Schattschneier [17] foun an elementary proof for the property that group Iso 1 (R 2 ) is the semi-irect prouct of D 4 an T (2), where 1 is the taxicab metric efine by (1.2) (for p = 1 an n = 2) an D 4 an T (2) are the symmetry group of the square an the translations group of R 2, respectively. A similar result hols for the group Iso 1 (R 3 ), i.e. this group is isomorphic to the semi-irect prouct of groups D h an T (3), where D h is the symmetry group of the Eucliean octaheron an T (3) is the translations group of R 3. This was recently prove by O. Gelisgen,

2 446 Vasile Bulgărean R. Kaya [7]. In fact the taxicab metric generates many interesting non-eucliean geometric properties (see the book of E.F. Krause [10] or the thesis of G. Chen [4]). Another result concerning the isometry group of the plane R 2 with respect to the Chinese checker metric C, where C (x, y) = max{ x 1 x 2, y 1 y 2 } + ( 2 1) min{ x 1 x 2, y 1 y 2 }, (1.1) was recently obtaine by R. Kaya, O. Gelisgen, S. Ekmekci, A. Bayar [9].They have showe that this group is isomorphic to the semi-irect prouct the Diheral group D 8, the Eucliean symmetry group of the regular octagon an T (2). In the paper [1] we have consiere X = R n, an for any real number p 1 the metric p efine by ( n ) 1/p p (x, y) = x i y i p, (1.2) i=1 where x = (x 1,..., x n ), y = (y 1,..., y n ) R n. If p =, then the metric is efine by (x, y) = max{ x 1 y 1,..., x n y n }. (1.3) In the case p = 2, we get the well-known Eucliean metric on R n. In this case we have the Ulam s Theorem which states that Iso 2 (R n ) is isomorphic to the semi-irect prouct of orthogonal group O(n) an T (n), where T (n) is the group of translations of R n (see for instance the references [5], [16]). The situation p 2 is very interesting. The main result of [1] consists in a complete escription of the groups Iso p (R n ) for p 1, p 2, an p =. We have prove that in case p 2 all these groups are isomorphic an consequently, they are not epening on number p. The main ingreient in the proof of the above result is the Mazur-Ulam Theorem about the isometries between norme linear spaces (see the original reference [12], the monograph [6], the references [14], [15] for some extensions, an [18], [19] for new proofs). In this paper we stuy some properties of the isometry group of the so calle French railroa metric. 2. The main results Let (X, ) be a metric space an let f : X X be a map. The minimal isplacement of f, enote by λ(f), is the greatest lower boun of the isplacement function of f, that is λ(f) = inf (x, f(x)). x X The minimal set of f, enote by Min(f), is the subset of X efine as Min(f) = {x X : (x, f(x)) = λ(f)}. Accoring to the monograph of A. Papaopoulos [13], we have the following general classification of the isometries of a metric space X in terms of the invariants λ(f) an Min(f). Consier f : X X to be an isometry. Then f is sai to be 1. Parabolic if Min(f) = Φ. 2. Elliptic if Min(f) Φ an λ(f) = 0. Thus, f is elliptic if an only if Fix(f) Φ.

3 The group of isometries of the French rail ways metric Hyperbolic if Min(f) Φ an λ(f) > 0. France is a centralize country: every train that goes from one French city to another has to pass through Paris. This is slightly exaggerate, but not too much. This motivates the name French railroa metric for the following construction. Let (X, ) be a metric space, an fix p X. Define a new metric F,p on X by letting { 0, if an only if x = y F,p (x, y) = (x, p) + (p, y), if x y (2.1) The French railroa metric generates a very poor geometry. In fact, the geometric properties are concentrate at the point p. For instance, consier X to be the plane R 2, p = O, the origin, an is the stanar Eucliean metric 2. Then for two istinct points A an B the perpenicular bisector of the segment [AB] exists if an only if AO = BO. Similarly, the Menger segment [AB] (see the monograph of W. Benz [3, p.43]) has a mipoint, if an only if AO = BO. In the case AO BO, the segment [AB] consists only in the points A an B. The main purpose of this paper is to erive some general properties of the group Iso F,p (X). We will give an equivalent property for Iso F,0 (R n ), the isometry group of the space R n with the fixe point the origin p = 0, an the Eucliean metric 2, an we will iscuss the complexity of an effective escription in the cases n = 1 an n = 2. Let Iso (p) (X) be the subgroup of Iso (X) consisting in all isometries fixing the point p, i.e. Iso (p) (X) = {h Iso (X) : h(p) = p}. Theorem 2.1. The following relation Iso (p) (X) Iso F,p (X) hols, that is Iso (p) (X) is a subgroup of Iso F,p (X). Proof. Let f Iso (p) (X). For every x, y X, x y, we have F,p (f(x), f(y)) = (f(x), p) + (p, f(y)) = (f(x), f(p)) + (f(p), f(y)) = (x, p) + (p, y) = F,p (x, y), hence, f belongs to Iso F,p (X). For x y, the relation F,p (f(x), f(y)) = F,p (x, y) is equivalent to (f(x), p) + (p, f(y)) = (x, p) + (p, y). (2.2) Theorem 2.2. For every isometry f Iso F,p (X), the point p is fixe, that is f(p) = p. Proof. Let start to investigate the topology generate by metric F,p. Consier the ball of raius r an centere in the point C, that is the set B F,p (C; r) = {x X : F,p (C, x) r}. The inequality F,p (C, x) r is equivalent to (C, p) + (p, x) r. If C p, then we obtain (p, x) r (C, p). This means that B F,p (C; r) = B (p; r (C, p) if (C, p) < r an B F,p (C; r) = {C} if (C, p) r. Therefore, the neighborhoos basis at the point p generate by F,p coincies with the the neighborhoos basis at the point p generate by the metric.

4 448 Vasile Bulgărean For every x, y X, x y, we have F,p (x, y) = (x, p) + (p, y) (x, y), hence Because f is a F,p -isometry, from (2.3) it follows Take y = p in relation (2.2) an obtain F,p (x, y) (x, y). (2.3) (f(x), f(y)) F,p (f(x), f(y)) = F,p (x, y). (f(x), p) + (p, f(p)) = (x, p). (2.4) Clearly, f is continuous in the topology generate by the metric F,p. Accoring to the result from the beginning of the proof, we have x p in the topology of F,p if an only if x p in the topology of. That is F,p (x, p) 0 if an only if (x, p) 0, hence the relation (2.4) implies (f(p), p) = 0, hence f(p) = p. From the result in Theorem 2.2, it follows : Corollary 2.3. For every metric space (X, ) an for every point p X, the metric space (X, F,p ) is of elliptic type, that is every its isometry is elliptic. 3. Comments on the case X = R n an = 2 Let consier X = R n with the Eucliean metric = 2, an the point p to be the origin of R n. If f is an isometry with respect to the inuce French railroa metric, then f satisfies relation (2.4) hence, for every x, y R n, we have f(x) + f(y) = x + y. (3.1) Accoring to Theorem 2.2, the origin of R n is a fixe point of f, i.e. we have f(0) = 0. Therefore, the relation (3.1) is equivalent to f(x) = x, x R n. (3.2) Therefore, f Iso F,0 (R n ) if an only if it is bijective an it satisfies the functional equation (3.2). The equation (3.2) ensures the continuity of f only at the point 0. For the sake of simplicity, let us enote by G n the isometry group Iso F,0 (R n ). If f is linear, then the equation (3.2) means that it is an orthogonal transformation of R n. It follows that the real orthogonal group O(n, R) is a subgroup of G n. The group { 1 R n, 1 R n} is a normal subgroup of G n. Also, if we impose some smoothness conitions we obtain interesting situations. Denote by G k n, k = 0, 1,,, the subgroup of G n consisting in all F,0 -isometries of class C k. Clearly, we get G n G k n... G 1 n G 0 n G n. On the other han, if f G k n, then the restriction f S n 1 is an element of the group Diff k (S n 1 ) of the C k - iffeomorphisms of the unity (n 1)-imensional sphere of the space R n. As we expect, a bijective extension of a C k - iffeomorphism h : S n 1 S n 1 to the space R n, preserving the property to satisfy (3.2), is not unique. This means that it is very possible that the group G k n is larger than Diff k (S n 1 ). In the case n = 1, the group G 1 is efine by all bijective maps f : R R, continuous at 0, an satisfying the functional equation f(x) = x, x R. The

5 The group of isometries of the French rail ways metric 449 relation f(x) = x is equivalent to f 2 (x) x 2 = 0, i.e. (f(x) x)(f(x) + x) = 0. Therefore, the general possible form of the maps in G 1 is { x if x R \ A f A (x) = x if x A where A is an arbitrary subset of R. For all real number a 0, the map x if x a, a f a (x) = a if x = a a if x = a belongs to G 1. Moreover, we have f a f a = 1 R hence {1 R, f a } is a subgroup of G 1. This means that the group G 1 has infinitely many subgroups isomorphic to Z 2. Another subgroup isomorphic to Z 2 is G 1 = { 1 R, 1 R }, an it contains all smooth iffeomorphisms of the real line satisfying the functional equation f(x) = x, x R. It is a normal subgroup of G 1. Clearly, 1 R preserves the orientation of the 0- imensional sphere S 0 = { 1, 1}, an 1 R reverses the orientation of S 0 = { 1, 1}. In the case n = 2, we can ientify R 2 with the complex plane C, an then the group G 2 is efine by all bijective maps f : C C, continuous at 0, an satisfying the functional equation f(z) = z, z C. The orthogonal group O(2, R) is the symmetry group of the circle S 1 = {z C : z = 1}. It is isomorphic (as a real Lie group) to the circle group (S 1, ), also known as U(1). In this case, the circle group (S 1, ) is a subgroup of G 2. Consequently, all subgroups of (S 1, ) are subgroups of G 2. References [1] Anrica, D., Bulgărean, V., Some Remarks on the Group of Isometries of a Metric Space, in Nonlinear Analysis (Stability, Approximation, an Inequalities), In honor of Th.M.Rassias on the occassion of his 60th birthay, P.M. Paralos, P.G. Georgiev, H.M. Srivasava, Es., Springer, 2012, [2] Anrica, D., Wiesler, H., On the isometry groups of a metric space (Romanian), Seminar Diactica Matematicii, 5(1989), 1-4. [3] Benz, W., Classical Geometries in Moern Contexts.Geometry of Real Inner Prouct Spaces, Secon Eition, Birkhäuser, Basel-Boston-Berlin, [4] Chen, G., Lines an Circles in Taxicab Geometry, Master Thesis, Department of Mathematics an Computer Science, Central Missouri State University, [5] Clayton, W.D., Eucliean Geometry an Transformations, Aison-Wesley, Reaing, MA, [6] Fleming, R.J., Jamison, J.E., Isometries on Banach Spaces: Function Spaces, Chapman Hall / CRC, Monographs an Surveys in Pure an Applie Mathematics No.129, Boca Raton, Lonon, New York, Washington D.C., [7] Gelisgen, O., Kaya, R., The taxicab space group, Acta Mathematica Hungarica, 122(2009), No. 1-2, [8] Kaya, R., Area Formula For Taxicab Triangles, Pi Mu Epsilon, 12(2006),

6 450 Vasile Bulgărean [9] Kaya, R., Gelisgen, O., Ekmekci, S., Bayar, A., On the group of the isometries of the plane with generalize absolute metric, Rocky Mountain Journal of Mathematics, 39(2009), no. 2. [10] Krause, E.F., Taxicab Geometry, Aison-Wesley Publishing Company, Menlo Park, CA, [11] Ozcan, M., Kaya, R., Area of a Triangle in Terms of the Taxicab Distance, Missouri J. of Math. Sci., 15(2003), [12] Mazur, S., Ulam, S., Sur les transformationes isometriques espaces vectoriels normes, C. R. Aca. Sci. Paris, 194(1932), [13] Papaopoulos, A., Metric Spaces, Convexity an Nonpositive Curvature, European Mathematical Society, [14] Park, C.-G., Rassias, Th.M., Isometries on linear n-norme spaces, Journal of Inequalities in Pure an Applie Mathematics, 7(5)(2006), Article 168, 7 pp. [15] Park, C.-G., Rassias, Th.M., Aitive isometries on Banach spaces, Nonlinear Functional Analysis an Applications, 11(5)(2006), [16] Richar, S.M., George, D.P., Geometry. A Metric Approach with Moels, Springer- Verlag, New York, [17] Schattschneier, D.J., The Taxicab Group, Amer. Math. Monthly, 91(1984), [18] Väisälä, J., A proof of the Mazur-Ulam theorem, Amer. Math. Monthly, bf 110(7)(2003), [19] Vogt, A., Maps which preserve equality of istance, Stuia Math., 45(1973), [20] Willar Jr., M., Symmetry Groups an Their Applications, Acaemic Press, New York, Vasile Bulgărean Babeş-Bolyai University Faculty of Mathematics an Computer Science Cluj-Napoca, Romania vasilebulgarean@yahoo.com