On the Measurability of Sets of Pairs of Intersecting Nonisotropic Straight Lines of Type Beta in the Simply Isotropic Space

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1 Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica 48 (2009) 7 6 On the Measurability of Sets of Pairs of Intersecting Nonisotropic Straight Lines of Type Beta in the Simply Isotropic Space Adrijan Varbanov BORISOV Margarita Georgieva SPIROVA 2 Department of Mathematics South-West University Neofit Rilski 66 Ivan Mihailov Str Blagoevgrad Bulgaria adribor@aix.swu.bg 2 Fakultät für Mathematik TU Chemnitz D-0907 Chemnitz Germany margarita.spirova@mathematik.tu-chemnitz.de (Received October ) Abstract The measurable sets of pairs of intersecting non-isotropic straight lines of type β and the corresponding densities with respect to the group of general similitudes and some its subgroups are described. Also some Croftontype formulas are presented. Key words: Simply isotropic space density measurability Mathematics Subject Classification: C6 Introduction The simply isotropic space I () (see [8]) is defined as a projective space P (R) in which the absolute consists of a plane ω (the absolute plane) and two complex conjugate straight lines f f 2 (the absolute lines) withinω. In homogeneous coordinates (x 0 x x 2 x ) we can choose the plane x 0 =0as the plane ω the line x 0 =0 x + ix 2 =0as the line f and the line x 0 =0 x ix 2 =0 as the line f 2. Then the intersecting point F of f and f 2 which is called an absolute point has coordinates (0 0 0 ). All regular projectivities transforming the absolute figure into itself form the 8-parametric group G 8 of general simply 7

2 8 Adrijan V. BORISOV Margarita G. SPIROVA isotropic similitudes. Inaffinecoordinates (x y z) with respect to the affine coordinate system (O e e 2 e ) any similitude of G 8 can be written in the form ([8 p. ]) x = c + c 7 (x cos ϕ y sin ϕ) y = c 2 + c 7 (x sin ϕ + y cos ϕ) z = c + c 4 x + c y + c 6 z () where c c 2 c c 4 c c 6 c 7 and ϕ are real parameters and c 7 > 0. A plane in I () is said to be non-isotropic if its infinite line is not incident with the absolute point F ; otherwise the plane is called isotropic. A straight line in I () is said to be (completely) isotropic if its infinite point coincides with the absolute point F ; otherwise the straight line is said to be non-isotropic ([8 p. ]). Let G and G 2 be two non-isotropic straight lines and let us denote by U and U 2 their infinite points respectively. The straight lines G and G 2 are said to be of type β if the points U U 2 andf are collinear; otherwise the straight lines are said to be of type α ([8 p. 4]). We will consider also the following subgroups of G 8 : I. B 7 G 8 c 7 =. This is the group of simply isotropic similitudes of the δ-distance ([8 p. ]). II. S 7 G 8 c 6 =. This is the group of simply isotropic similitudes of the s-distance ([8 p. 6]). III. W 7 G 8 c 6 = c 7. This is the group of simply isotropic angular similitudes ([8 p. 8]). IV. G 7 G 8 ϕ =0. This is the group of simply isotropic boundary similitudes ([8 p. 8]). V. V 7 G 8 c 6 c 2 7 =. This is the group of simply isotropic volume preserving similitudes ([8 p. 8]). VI. G 6 = G 7 V 7. This is the group of simply isotropic volume preserving boundary similitudes ([8 p. 8]). VII. B 6 = B 7 G 7. Thisisthegroupofmodular boundary motions ([8 p. 9]). VIII. B = B 7 S 7 G 7. Thisisthegroupofunimodular boundary motions ([8 p. 9]). Basic references on the geometry of the simply isotropic space I () are Sachs book [8] and Strubecker s papers [8] [] and [2]. Using some basic concepts from integral geometry in the sense of R. Deltheil [] M. I. Stoka [0] G. I. Drinfel d and A. V. Lucenko [4] [] [6] we study the measurability of sets of pairs of intersecting nonisotropic straight lines of type β with respect to G 8 and indicated above subgroups. Analogous problems about sets of pairs of intersecting non-isotropic straight lines of type α in I () have been treated in [2].

3 On the measurability of sets of pairs Measurability with respect to G 8 Let (G G 2 ) be a pair of intersecting non-isotropic straight lines of type β. Let G i have Plücker coordinates (p i j ) i = 2 j =...6 which satisfy the relations ([8 p. 8]) p i pi 4 + pi 2 pi + pi pi 6 =0 i = 2. (2) Since G and G 2 are intersecting non-isotropic lines of type β wehave p p p 2p 2 + p p p 4p 2 + p p p 6p 2 =0 p p 2 0 () p i + pi 2 0 i = 2 (4) p p2 2 p 2 p2 =0. () Having in mind (4) we can assume without loss of generality that p i =. From (2) p i 4 can be expressed by the remaining Plücker coordinates of G iand in view if () and () p 2 2 and p 2 6 also can be expressed by p 2 p p p 6 p 2 and p 2. Thus the pair (G G 2 ) can be determined by p 2 p p p 6 p2 p2. Remark 2. We note that if G i i = 2 are represented in the usual way by the equations G : { x = a (z r)+p y = b (z r)+q G 2 : { x = a2 (z r)+p y = a 2 b (z r)+q (6) a where P (p q r) =G G 2 and a 0 a 2 0then p 2 = b a p = a p = r p a p 6 = p b a q p 2 = a 2 p 2 = r p a 2. (7) Under the action of () the pair (G G 2 )(p 2p p p 6p 2 p 2 ) is transformed into the pair (G G 2 )(p 2 p p p 6 p2 p2 ).Thuswehave p 2 = Kc 7 (sin ϕ + p 2 cos ϕ) p = K(c 4 + c p 2 + c 6p ) p = K{(c c p 6 + c 6 p )c 7 cos ϕ [c + c 4 p 6 + c 6(p 2 p + p p 6 )]c 7 sin ϕ c (c 4 + c + c 6 p )} p 6 = Kc 7[(c p 2 c 2)cosϕ +(c + c 2 p 2 )sinϕ + c 7p 6 ] p 2 = K(c 4 + c p 2 + c 6p 2 ) p 2 = K{(c c p 6 + c 6p 2 )c 7 cos ϕ [c + c 4 p 6 + c 6(p 2 p2 + p2 p 6 )]c 7 sin ϕ c (c 4 + c + c 6 p 2 )} (8)

4 0 Adrijan V. BORISOV Margarita G. SPIROVA where K =[c 7 (cos ϕ p 2 sin ϕ)] i = 2. The transformations (8) form the associated group G 8 of G 8 ([0 p. 4]). The group G 8 is isomorphic to G 8 and the density with respect to G 8 of the pairs (G G 2 ) if it exists coincides with the density with respect to G 8 of the set of parameters (p 2 p p p 6 p2 p2 ). The associated group G 8 has the infinitesimal operators X = p p p 2 p 6 p 2 p 2 X 2 = p X = 6 p + p 2 X 4 = p + p 2 X = p 2 p p 6 p + p 2 p 2 p 6 p 2 X 6 = p p + p p + p 2 p 2 + p 2 p 2 X 7 = p p p 6 p 6 + p 2 X 8 =[+(p 2 )2 ] p + p 2 p 2 p p p 6 p + p 2 p 6 p + p 2 p2 6 p 2 g 6 p2 p 2 (9) and it acts transitively on the set of parameters (p 2p p p 6p 2 p 2 ). The infinitesimal operators X X 2 X X 4 X 7 andx 8 are arcwise unconnected and X 6 = p2 p p 2 X + p p 6 X 2 + p p 2 p p 2 p 2 X + X 7. p Since X ( p2 p p 2 )+X 2 (p 6)+X ( p p2 p p2 p p 2 )+X 7 () = 0 p we can establish the following Theorem 2. The set of pairs of intersecting non-isotropic straight lines is not measurable with respect to the group G 8 and it has no measurable subsets. p 2 Measurability with respect to S 7 The associated group S 7 of the group S 7 has the infinitesimal operators X X 2 X X 4 X X 7 andx 8 from (9) and it acts transitively on the set of parameters (p 2 p p p 6 p2 p2 ). The integral invariant function f = f(p 2p p p 6p 2 p 2 ) satisfying the so-called system of R. Deltheil (see [ p. 28]; [0 p. ]) X (f) =0 X 2 (f) =0 X (f) =0 X 4 (f) =0 X (f) =0 X 7 (f)+f =0 X 8 (f)+p 2 f =0 has the form where h = const. f = h (p p2 )[ + (p 2 )2 ] 2

5 On the measurability of sets of pairs... Thus we state the following Theorem. The set of pairs (G G 2 )(p 2 p p p 6 p2 p2 ) is measurable with respect to the group S 7 and has the density p 2 p [ + (p 2 )2 ] 2 dp 2 dp dp dp 6 dp2 dp2. (0) Differentiating (7) and substituting into (0) we obtain other expression for the density: Corollary. The density (0) for the pairs (G G 2 ) represented by (6) can be written in the form a a 2 2 (a2 + da b2 )2 db da 2 dp dq dr. () 4 Some Crofton-type formulas with respect to S 7 Let us consider the isotropic plane ι which is determined by the lines G and G 2. The plane ι has the equation ι: b x a y + a q b p =0. If P is the orthogonal projection of P into Oxy consider the affine coordinate system ( P e e2 ) in the isotropic plane ι where e =(a b ) e 2 = e. It should be noticed that if G = ι Oxy then e G. Let J = Oxz ι and J 2 = Oyz ι. Obviously J : x = p a q y =0 J 2 : y = q b p x =0 b a and J J 2 have the equations J : x = q J 2 : x = p b a with respect to ( P e e2 ). Then the density d(j J 2 ) for the pairs (J J 2 ) with respect to the group H4 which is the restriction of S 7 into ι is (see [ p. 20]) ( p d(j J 2 )= q ) 2 d p d q. a b a b Recall that ([8 p. 4]) s = a a 2 (2) a 2 a 2 + b 2 is the angle from G to G 2 we find d(j J 2 ) dp ds = (pb qa )pq da a b4 2 a2 a 2 + b 2 db dp dq dr da 2.

6 2 Adrijan V. BORISOV Margarita G. SPIROVA Comparing with () we get a 4 b4 pq(pb qa )(a 2 + b2 ) 2 d(j J 2 ) ds dp. () Let ϕ i i = 2 b the angle between G i and Oxy. Then([8p.48]) a ϕ = ϕ a 2 + b 2 2 = (4) a 2 a 2 + b 2 and () becomes a 4 b 4 ϕ pq(pb qa ) d(j J 2 ) ds dp. () By differentiation of (4) and by exterior multiplication by (2) we obtain a 4 b4 pq(pb qa )(a 2 + b2 ) 2 d(j J 2 ) dϕ 2 dp = a 4 b 4 ϕ pq(pb qa ) d(j J 2 ) dϕ 2 dp. (6) If ϕ is the isotropic distance from J to J 2 then([7p.9]) ϕ = p a + q b. (7) Putting (7) into () and (6) we find a b ϕ pq ϕ d(j J 2 ) ds dp = a b ϕ pq ϕ d(j J 2 ) dϕ 2 dp. (8) Let G i and G2 i be now the projections of G i into Oxz and Oyz obtained in a parallel way to Oy and Ox respectively. Then G i : z = a i x + r p a i y=0i= 2 G 2 : z = b y + r q b x=0 Furthermore G 2 2 : z = a a 2 b y + r a a 2 b q x =0. d(g G 2 )= a a 2 (a 2 a ) da da 2 dp dr (9) is the density for the pairs (G G 2 ) in the isotropic plane Oxz with respect H4 which is the restriction of S 7 into Oxz and d(g 2 G 2 2)= b 2 a 2(a 2 a ) (a db da 2 a 2 db da ) dq dr

7 On the measurability of sets of pairs... is the density for the pairs (G 2 G2 2 ) in the isotropic plane Oyz with respect 2 H4 which is the restriction of S 7 into Oyz (see [ p. 77]). By exterior multiplication of (G G 2 ) and ds dq weget a 2 ϕ b d(g G 2) ds dq (20) and by exterior multiplication of (9) and dϕ dq: a 2 s b d(g G 2 ) dϕ dq. (2) If instead of using dϕ dq we multiply by dϕ 2 dq weobtain a a 2 s b d(g G 2 ) dϕ 2 dq. (22) Analogously we can derive the following formulas: a 2 b 2 ϕ a d(g2 G 2 2) ds dp 2 = b 2 s a d(g2 G 2 2) dϕ dp = a 2 b 2 s a 2 d(g2 G 2 2) dϕ 2 dp. (2) In summary the following theorem holds. Theorem 4. The density for the set of pairs (G G 2 ) of intersecting nonisotropic straight lines of type β determined by (6) with respect to the group S 7 satisfies the relations () (6) (8) (20) (2) (22) and (2). Measurability with respect to G 6 Now the corresponding associated group G 6 has the infinitesimal operators Y = p Y = p p p 2 + p 2 p 6 + p 2 Y 4 = p 2 p 2 p Y 2 = p 6 p p 6 + p 2 p 2 p 6 Y 7 =p p +2p p p 6 p +p 2 6 p 2 +2p 2 p 2 Y 8 = p + p 2. The group G 6 acts intransitively on the set of points (p 2p p p 6p 2 p 2 ) and therefore the set of pairs (G G 2 ) has not invariant density with respect to G 6. The system p 2 Y (f) =0Y 2 (f) =0Y (f) =0Y 4 (f) =0Y 7 (f) =0Y 8 (f) =0

8 4 Adrijan V. BORISOV Margarita G. SPIROVA has the solution f = p 2 and it is an absolute invariant of G 6. Consider the subset of pairs (G G 2 ) satisfying the condition p 2 = h (24) where h = const. The group G 6 induces on this subset the group G 6 with the infinitesimal operators Z = p Z = p h p + p Z 4 = p 2 p p 2 p 2 Z 2 = p p 6 p 6 p + p 2 p 2 p 6 Z 7 =p p +2p p p 6 p +p 2 6 p 2 +2p 2 p 2 Z 8 = p + p 2. The integral invariant function f = f(p p p 6 p2 p2 ) which satisfies the Deltheil system Z (f) =0 Z 2 (f) =0Z (f) =0Z 4 (f) =0Z 7 (f) 9f =0Z 8 (f) =0 has the form where c = const. Thus we state the following f = c (p p2 ) Theorem. The set of pairs (G G 2 )(p 2p p p 6p 2 p 2 ) of intersecting nonisotropic lines of type β is not measurable with respect to G 6 but it has the measurable subset p 2 = h h = const with the density p 2 p 2 p dp dp dp 6 dp 2 dp 2. (2) Differentiating (7) (24) and replacing into (2) we establish Corollary. The set of pairs (G G 2 ) of intersecting non-isotropic lines of type β determined by (6) is not measurable with respect to the group G 6 but ithasthemeasurablesubset b = h h = const a with the density (a a 2 ) 2 da da 2 dp dq dr.

9 On the measurability of sets of pairs... 6 Measurability with respect to B 7 W 7 G 7 V 7 B 6 and B By arguments similar to those used in the sections 2 and we investigated the measurability with respect to all the remaining groups. We have the following results: Theorem 6. The set of pairs (G G 2 ) of intersecting non-isotropic straight lines of type β determined by (6) is measurable with respect to the group (i) B 7 and it has the density a a 2 (a a 2 ) a 2 + b2 da db da 2 dp dq dr; (ii) V 7 and it has the density a (a a 2 ) 2 (a 2 + b2 ) da db da 2 dp dq dr. Theorem 6.2 With respect to the groups W 7 and S 7 the set of pairs (G G 2 ) of intersecting non-isotropic lines of type β is not measurable and it has no measurable subsets. Theorem 6. The set of pairs (G G 2 ) of intersecting non-isotropic straight lines of type β determined by (6) is not measurable with respect to the group (i) B 6 butithasthemeasurablesubset b a = h h = const with the density a a 2 (a a 2 ) da da 2 dp dq dr; (ii) B but it has the measurable subset b a = h a a 2 = h 2 h h 2 = const with the density a 2 a (a a 2 ) da da 2 dp dq dr.

10 6 Adrijan V. BORISOV Margarita G. SPIROVA References [] Borisov A.: Integral geometry in the Galilean plane. Research work qualifying for a degree full-professor Sofia 998. [2] Borisov A. V. Spirova M. G.: Crofton-type formulas relating sets of pairs of intersecting nonisotropic straight lines in the simply isotropic space. Tensor 67 (2006) [] Deltheil R.: Sur la théorie des probabilité géométriques. Thése Ann. Fac. Sc. Univ. Toulouse (99) 6. [4] Drinfel d G. I.: On the measure of the Lie groups. Zap. Mat. Otdel. Fiz. Mat. Fak. Kharkov. Mat. Obsc. 2 (949) 47 7 (in Russian). [] Drinfel d G. I. Lucenko A. V.: On the measure of sets of geometric elements. Vest. Kharkov. Univ. (964) 4 4 (in Russian). [6] Lucenko A. V.: On the measure of sets of geometric elements and thear subsets. Ukrain. Geom. Sb. (96) 9 7 (in Russian). [7] Sachs H.: Ebene isotrope Geometrie. Friedr. Vieweg Sohn Braunschweig 987. [8] Sachs H.: Isotrope Geometrie des Raumes. Friedr. Vieweg and Sohn Braunschweig Wiesbaden 990. [9] Santaló L. A.: Integral Geometry and Geometric Probability. Addison-Wesley London 976. [0] Stoka M. I.: Geometrie Integrala. Ed. Acad. RPR Bucuresti 967. [] Strubecker K.: Differentialgeometrie des isotropen Raumes I. Sitzungsber. Österr. Akad. Wiss. Wien 0 (94). [2] Strubecker K.: Differentialgeometrie des isotropen Raumes II III IV V. Math. Z. 47 (942) ; 48 (942) ; 0 (944) 92; 2 (949) 2 7.