On geometry of gonometric family of cycles

On geometry of gonometric family of cycles V aleri i Dryuma ∗ Institute of Mathem atics and Informatics, AS RM, 5 A c ademiei Str e et, 20 2 8 Kishinev, Moldova , e-mail: v a lery@dryuma.c o m ; c ainar@mail. m d Abstract An examples solutions of t he equation for curv ature of congruence of cycles are considered. Their prop erties are discussed. 1 Gonometric family of cyc les Tw o parametrical family o f cycles on the plane is determined b y the equation ( ξ − x ) 2 + ( η − y ) 2 − φ ( x, y ) 2 = 0 . The angle metric in a given family of cycles is defined by the expression [1] ds 2 =  1 −  ∂ ∂ x φ ( x, y )  2  dx 2 − 2 ∂ ∂ x φ ( x, y ) ∂ ∂ y φ ( x, y ) dx dy +  1 −  ∂ ∂ y φ ( x, y )  2  dy 2 ( φ ( x, y )) 2 . (1) The expression for the curv at ur e of the metric (1 ) has t he form K ( x, y ) = φ  1 −  ∂ ∂ y φ  2  ∂ 2 ∂ x 2 φ + 2  ∂ ∂ y φ   ∂ ∂ x φ  ∂ 2 ∂ x∂ y φ +  1 −  ∂ ∂ x φ  2  ∂ 2 ∂ y 2 φ   1 −  ∂ ∂ x φ  2 −  ∂ ∂ y φ  2  2 − − φ 2   ∂ 2 ∂ x 2 φ  ∂ 2 ∂ y 2 φ −  ∂ 2 ∂ x∂ y φ  2   1 −  ∂ ∂ x φ  2 −  ∂ ∂ y φ  2  2 −   1 − ∂ ∂ x φ ! 2 − ∂ ∂ y φ ! 2   − 1 + 1 . (2) ∗ W ork supp orted in par t by Gran t RFFI, Russia- Moldov a 1 2 The con gruence of cycles of co nstan t curv ature The congruence o f cycles of constant curv ature K ( x, y ) = K are defined by the Monge-Amp ere equation [1] K = φ  1 −  ∂ ∂ y φ  2  ∂ 2 ∂ x 2 φ + 2  ∂ ∂ y φ   ∂ ∂ x φ  ∂ 2 ∂ x∂ y φ +  1 −  ∂ ∂ x φ  2  ∂ 2 ∂ y 2 φ   1 −  ∂ ∂ x φ  2 −  ∂ ∂ y φ  2  2 − − φ 2   ∂ 2 ∂ x 2 φ  ∂ 2 ∂ y 2 φ −  ∂ 2 ∂ x∂ y φ  2   1 −  ∂ ∂ x φ  2 −  ∂ ∂ y φ  2  2 −   1 − ∂ ∂ x φ ! 2 − ∂ ∂ y φ ! 2   − 1 + 1 . (3) 2.1 Metho d of so lution F or solutions of the Monge-Amp ere equation F ( x, y , f x , f y , f xx , f xy , f y y ) = 0 w e use the metho d of solution o f the p.d.e.’s describ ed first in [2] and dev elop ed la ter in [3 ] . This method allo w us to construct particular s olutions of the par tial nonlinear differen tial equation F ( x, y , z , f x , f y , f z , f xx , f xy , f xz , f y y , f y z , f xxx , f xy y , f xxy , .. ) = 0 (4) with the help o f transformatio n of the function and corresp onding v ariables. Essence of the metho d consists in a fo llowing presen t a tion of the functions and v ariables f ( x, y , z ) → u ( x, t, z ) , y → v ( x, t, z ) , f x → u x − u t v t v x , f z → u z − u t v t v z , f y → u t v t , f y y → ( u t v t ) t v t , f xy → ( u x − u t v t v x ) t v t , ... (5) where v aria ble t is considered as parameter. Remark that conditions of the type f xy = f y x , f xz = f z x ... are fulfilled at the suc h ty p e of presen t a tion. In result instead of equation (4) one get the relation b etw een the new v ariables u ( x, t, z ), v ( x, t, z ) and their pa rtial deriv ativ es Ψ( u, v , u x , u z , u t , v x , v z , v t ... ) = 0 . (6) This relatio n coincides with initial p.d.e at the condition v ( x, t, z ) = t and lead to the new p.d.e Φ( ω , ω x , ω t , ω xx , ω xt , ω tt , ... ) = 0 (7) when the functions u ( x, t, s ) = F ( ω ( x, t, z ) , ω t ... ) and v ( x, t, s ) = Φ( ω ( x, t, z ) , ω t ... ) are expressed through the auxiliary function ω ( x, t, s ). Remark that there are a v arious means to reduce the relation (6) in to the par t ia l differen tial equation. In a some cases the solution of equation (7) is a more simple problem than solution of equation (4). 2 Remark 1 As the examp le we c onsid e r the Monge-Amp er e e quation ∂ 2 ∂ x 2 f ( x, y ) ! ∂ 2 ∂ y 2 f ( x, y ) − ∂ 2 ∂ x∂ y f ( x, y ) ! 2 + 1 = 0 . After the ( u , v ) -tr ansform ation v ( x, t ) = ∂ ∂ t ω ( x, t ) ! t − ω ( x, t ) , u ( x, t ) = ∂ ∂ t ω ( x, t ) ! it takes the form of line ar e quation t 4 ∂ 2 ∂ t 2 ω ( x, t ) − ∂ 2 ∂ x 2 ω ( x, t ) = 0 . with gener al solution ω ( x, t ) = t  F1 ( − tx − 1 t ) + F2 ( tx + 1 t )  , dep end i n g fr om two arbitr ary functions. Choic e of the functions Fi and elimination of the p ar ameter t fr om c orr esp onding r elations le ad to the f unction f ( x, y ) satisfying the Monge- Amp er e e quation. 2.2 Congruence of cycles of zero curv ature In the case K = 0 we get the equation φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) − φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ ∂ y φ ( x, y ) ! 2 + 2 φ ( x, y ) ∂ ∂ y φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! ∂ 2 ∂ x∂ y φ ( x, y ) + φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) − − φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! 2 − ( φ ( x, y )) 2 ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ 2 ∂ y 2 φ ( x, y )+ + ( φ ( x, y )) 2 ∂ 2 ∂ x∂ y φ ( x, y ) ! 2 − ∂ ∂ x φ ( x, y ) ! 2 − ∂ ∂ y φ ( x, y ) ! 2 + + ∂ ∂ x φ ( x, y ) ! 4 + 2 ∂ ∂ x φ ( x, y ) ! 2 ∂ ∂ y φ ( x, y ) ! 2 + ∂ ∂ y φ ( x, y ) ! 4 = 0 (8) After applying o f the (u,v)-transformat io n at this equation is r educed to the relation ∂ ∂ t u ( x, t ) ! 4 − 4 ∂ ∂ x u ( x, t ) ! ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! 3 ∂ ∂ x v ( x, t ) ! 3 − − 4 ∂ ∂ x u ( x, t ) ! 3 ∂ ∂ t v ( x, t ) ! 3 ∂ ∂ t u ( x, t ) ! ∂ ∂ x v ( x, t )+ 3 +6 ∂ ∂ x u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! 2 ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ x v ( x, t ) ! 2 − − u ( x, t ) ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! 2 ∂ 2 ∂ x 2 u ( x, t ) + u ( x, t ) ∂ ∂ t v ( x, t ) ! 2 ∂ 2 ∂ t 2 u ( x, t ) ! ∂ ∂ x v ( x, t ) ! 2 − − ( u ( x, t )) 2 ∂ ∂ t u ( x, t ) ! 2 ∂ 2 ∂ x 2 v ( x, t ) ! ∂ 2 ∂ t 2 v ( x, t ) + 2 ∂ ∂ t u ( x, t ) ! 4 ∂ ∂ x v ( x, t ) ! 2 − − ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! 2 − 2 ( u ( x, t )) 2 ∂ 2 ∂ t∂ x u ( x, t ) ! ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ t∂ x v ( x, t )+ + ( u ( x, t )) 2 ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ x 2 v ( x, t ) ! ∂ ∂ t v ( x, t ) ! ∂ 2 ∂ t 2 u ( x, t )+ + ( u ( x, t )) 2 ∂ 2 ∂ x 2 u ( x, t ) ! ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ t 2 v ( x, t )+( u ( x, t )) 2 ∂ ∂ t u ( x, t ) ! 2 ∂ 2 ∂ t∂ x v ( x, t ) ! 2 − − u ( x, t ) ∂ ∂ t v ( x, t ) ! 2 ∂ 2 ∂ t 2 u ( x, t ) ! ∂ ∂ x u ( x, t ) ! 2 − u ( x, t ) ∂ ∂ t v ( x, t ) ! 3 ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ x 2 v ( x, t ) − − 2 u ( x, t ) ∂ ∂ t v ( x, t ) ! 3 ∂ 2 ∂ t∂ x u ( x, t ) ! ∂ ∂ x v ( x, t ) − 4 ∂ ∂ t u ( x, t ) ! 3 ∂ ∂ t v ( x, t ) ! ∂ ∂ x u ( x, t ) ! ∂ ∂ x v ( x, t ) − − u ( x, t ) ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ t 2 v ( x, t ) − u ( x, t ) ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ ∂ x v ( x, t ) ! 2 ∂ 2 ∂ t 2 v ( x, t )+ +2 u ( x, t ) ∂ ∂ t u ( x, t ) ! ∂ ∂ t v ( x, t ) ! 2 ∂ ∂ x u ( x, t ) ! ∂ 2 ∂ t∂ x u ( x, t )+ +2 u ( x, t ) ∂ ∂ t v ( x, t ) ! 2 ∂ ∂ t u ( x, t ) ! ∂ ∂ x v ( x, t ) ! ∂ 2 ∂ t∂ x v ( x, t ) − − 2 u ( x, t ) ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! ∂ ∂ x u ( x, t ) ! ∂ 2 ∂ t∂ x v ( x, t )+ + u ( x, t ) ∂ ∂ t v ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ 2 ∂ t 2 v ( x, t ) ! ∂ ∂ x u ( x, t ) ! 2 − − ( u ( x, t )) 2 ∂ 2 ∂ x 2 u ( x, t ) ! ∂ ∂ t v ( x, t ) ! 2 ∂ 2 ∂ t 2 u ( x, t )+2 ∂ ∂ t v ( x, t ) ! 3 ∂ ∂ x u ( x, t ) ! ∂ ∂ t u ( x, t ) ! ∂ ∂ x v ( x, t )+ + u ( x, t ) ∂ ∂ t u ( x, t ) ! 3 ∂ ∂ t v ( x, t ) ! ∂ 2 ∂ x 2 v ( x, t ) + ∂ ∂ t u ( x, t ) ! 4 ∂ ∂ x v ( x, t ) ! 4 + + ∂ ∂ x u ( x, t ) ! 4 ∂ ∂ t v ( x, t ) ! 4 − ∂ ∂ t v ( x, t ) ! 4 ∂ ∂ x u ( x, t ) ! 2 − − ∂ ∂ t v ( x, t ) ! 2 ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ x v ( x, t ) ! 2 + 2 ∂ ∂ t u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! 2 ∂ ∂ x u ( x, t ) ! 2 + 4 + u ( x, t ) ∂ ∂ t v ( x, t ) ! 2 ∂ 2 ∂ t 2 u ( x, t ) + u ( x, t ) ∂ ∂ t v ( x, t ) ! 4 ∂ 2 ∂ x 2 u ( x, t )+ + ( u ( x, t )) 2 ∂ 2 ∂ t∂ x u ( x, t ) ! 2 ∂ ∂ t v ( x, t ) ! 2 = 0 . F rom here after the choice of the functions u and v in the f o rm v ( x, t ) = t ∂ ∂ t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂ ∂ t ω ( x, t ) w e find the equation − ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 ∂ ∂ x ω ( x, t ) ! 2 − t ∂ ∂ t ω ( x, t ) − ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ x 2 ω ( x, t )+ + ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ x 2 ω ( x, t ) + ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 − − ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 − ∂ ∂ t ω ( x, t ) ! t ∂ ∂ x ω ( x, t ) ! 2 + + ∂ 2 ∂ t 2 ω ( x, t ) − ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 + ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ x ω ( x, t ) ! 4 + +2 ∂ ∂ x ω ( x, t ) ! 2 ∂ 2 ∂ t 2 ω ( x, t ) + 2 ∂ ∂ t ω ( x, t ) ! t 2 ∂ 2 ∂ t∂ x ω ( x, t ) ! ∂ ∂ x ω ( x, t )+ + ∂ ∂ t ω ( x, t ) ! 2 ∂ 2 ∂ x 2 ω ( x, t ) = 0 ha ving the particular solution ω ( x, t ) = A ( t ) + x, where − 2 d 2 dt 2 A ( t ) ! t 2 − 2 t d dt A ( t ) + 4 d 2 dt 2 A ( t ) = 0 . General solution of this equation is A ( t ) = C1 + C2 ln( t + √ t 2 − 2) No w elimination o f the v ar ia ble t from t he relations y √ t 2 − 2 − t C2 + C1 √ t 2 − 2 + C2 ln( t + √ t 2 − 2) √ t 2 − 2 + x √ t 2 − 2 = 0 , and φ ( x, y ) √ t 2 − 2 − C2 = 0 giv e us the function φ ( x, y ) defined fr o m the equation y − q 2 ( φ ( x, y ) ) 2 + 1 + C1 + ln( q 2 ( φ ( x, y ) ) 2 + 1 + 1 φ ( x, y ) ) + x = 0 satisfying the equation (8). 5 2.3 Congruence of p ositiv e constan t curv ature In the case K = 1 f rom (3) w e get the equation φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) − φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ ∂ y φ ( x, y ) ! 2 + +2 φ ( x, y ) ∂ ∂ y φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! ∂ 2 ∂ x∂ y φ ( x, y )+ + φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) − φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! 2 − − ( φ ( x, y )) 2 ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ 2 ∂ y 2 φ ( x, y ) + ( φ ( x, y )) 2 ∂ 2 ∂ x∂ y φ ( x, y ) ! 2 − 1+ + ∂ ∂ x φ ( x, y ) ! 2 + ∂ ∂ y φ ( x, y ) ! 2 = 0 . (9) The ( u, v )- transformation with condition v ( x, t ) = t ∂ ∂ t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂ ∂ t ω ( x, t ) lead to the equation ∂ ∂ t ω ( x, t ) ! 2 ∂ 2 ∂ x 2 ω ( x, t ) + ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 − ∂ 2 ∂ t 2 ω ( x, t ) ! t 4 + +2 ∂ ∂ t ω ( x, t ) ! t 2 ∂ 2 ∂ t∂ x ω ( x, t ) ! ∂ ∂ x ω ( x, t ) + ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ x 2 ω ( x, t ) − − ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ x 2 ω ( x, t ) − ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 + + ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 + ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 ∂ ∂ x ω ( x, t ) ! 2 − t ∂ ∂ t ω ( x, t ) − − ∂ ∂ t ω ( x, t ) ! t ∂ ∂ x ω ( x, t ) ! 2 = 0 . (10) This equation a dmits particular solution in the fo rm ω ( x, t ) = A ( t ) + x where the function A ( t ) is solution of equation 2 d 2 dt 2 A ( t ) ! t 2 − d 2 dt 2 A ( t ) ! t 4 − 2 t d dt A ( t ) = 0 . So w e get A ( t ) = C1 + C2 √ − 2 + t 2 6 No w after elimination of the parameter t from the r elat io ns y √ − 2 + t 2 + C1 √ − 2 + t 2 − 2 C2 + x √ − 2 + t 2 = 0 and φ ( x, y ) √ t 2 − 2 − C2 = 0 w e obtain the simplest solution o f the equation (9 ). φ ( x, y ) = 1 / 2 q 2 y 2 + 4 y C1 + 4 y x + 2 C1 2 + 4 C1 x + 2 x 2 + 4 C2 2 . Mo ore complicated solutions of the equation (1 0) in the form ω ( x, t ) = A ( t ) + B ( t ) x lead to the conditions B ( t ) = − √ t 2 − 1 , and A ( t ) is arbitrary function. In result elimination of the pa rameter t fro m the relations y √ t 2 − 1 − t d dt A ( t ) ! √ t 2 − 1 + A ( t ) √ t 2 − 1 + x = 0 , and φ ( x, y ) √ t 2 − 1 − d dt A ( t ) ! √ t 2 − 1 + tx = 0 with a giv en function A ( t ) w e get the solution of the equation (9 ) dep enden t from c hoice of arbitrary function. As example in the case A ( t ) = 1 t w e find the solutio n of the equation (9) in t he form 16 ( φ ( x, y )) 4 +  8 y 2 − 8 x 2 − 32  ( φ ( x, y )) 3 +  − 32 y 2 + 16 − 8 x 2 + x 4 + y 4 + 2 y 2 x 2  ( φ ( x, y )) 2 + +  8 y 2 − 10 y 4 + 8 x 4 − 2 y 2 x 2 + 32 x 2  φ ( x, y ) − y 6 − x 6 +20 y 2 x 2 − 8 x 4 − 3 y 2 x 4 − 16 x 2 − 3 y 4 x 2 + y 4 = 0 . 2.4 Congruence of negativ e constan t curv ature In the case K = − 1 fro m (3) we get the equation φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) − φ ( x, y ) ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ ∂ y φ ( x, y ) ! 2 + +2 φ ( x, y ) ∂ ∂ y φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! ∂ 2 ∂ x∂ y φ ( x, y ) + φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) − 7 − φ ( x, y ) ∂ 2 ∂ y 2 φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! 2 − ( φ ( x, y )) 2 ∂ 2 ∂ x 2 φ ( x, y ) ! ∂ 2 ∂ y 2 φ ( x, y )+ + ( φ ( x, y )) 2 ∂ 2 ∂ x∂ y φ ( x, y ) ! 2 + 1 − 3 ∂ ∂ x φ ( x, y ) ! 2 − 3 ∂ ∂ y φ ( x, y ) ! 2 + +2 ∂ ∂ x φ ( x, y ) ! 4 + 4 ∂ ∂ x φ ( x, y ) ! 2 ∂ ∂ y φ ( x, y ) ! 2 + 2 ∂ ∂ y φ ( x, y ) ! 4 = 0 . (11) The ( u, v )- transformation with condition u ( x, t ) = t ∂ ∂ t ω ( x, t ) − ω ( x, t ) , v ( x, t ) = ∂ ∂ t ω ( x, t ) lead to the equation ∂ ∂ t ω ( x, t ) ! 2 ∂ 2 ∂ x 2 ω ( x, t ) + ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 − ∂ 2 ∂ t 2 ω ( x, t ) ! t 4 + +2 ∂ ∂ t ω ( x, t ) ! t 2 ∂ 2 ∂ t∂ x ω ( x, t ) ! ∂ ∂ x ω ( x, t ) + ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ x 2 ω ( x, t ) − − ∂ 2 ∂ t 2 ω ( x, t ) ! ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ x 2 ω ( x, t ) − ∂ ∂ t ω ( x, t ) ! t 3 ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 + + ∂ ∂ t ω ( x, t ) ! t ∂ 2 ∂ t∂ x ω ( x, t ) ! 2 + ∂ 2 ∂ t 2 ω ( x, t ) ! t 2 ∂ ∂ x ω ( x, t ) ! 2 − t ∂ ∂ t ω ( x, t ) − − ∂ ∂ t ω ( x, t ) ! t ∂ ∂ x ω ( x, t ) ! 2 = 0 . (12) This equation a dmits the part icular solution ω ( x, t ) = √ − t 2 + 1 x + A ( t ) with arbitrary function A ( t ). In particular case after elimination of the parameter t from the relat io ns y √ − t 2 + 1 + xt − 2 t √ − t 2 + 1 = 0 and φ ( x, y ) √ − t 2 + 1 − t 2 √ − t 2 + 1 + x = 0 w e find the solutio n of the equation (11) in the form − 16 ( φ ( x, y )) 4 +  − 32 + 8 y 2 − 8 x 2  ( φ ( x, y )) 3 + +  − 2 y 2 x 2 + 32 y 2 − 16 − y 4 + 8 x 2 − x 4  ( φ ( x, y )) 2 + +  − 2 y 2 x 2 − 10 y 4 + 32 x 2 + 8 x 4 + 8 y 2  φ ( x, y ) − − y 4 + 16 x 2 + 3 x 4 y 2 + 8 x 4 + 3 y 4 x 2 + x 6 + y 6 − 20 y 2 x 2 = 0 8 3 Geo desic e quations The geo desic o f the metric (1) are equiv alen t to the equation d 2 dx 2 y ( x ) + a 1 ( x, y ) d dx y ( x ) ! 3 + 3 a 2 ( x, y ) d dx y ( x ) ! 2 + 3 a 3 ( x, y ) d dx y ( x ) + a 4 ( x, y ) = 0 , where a 1 ( x , y ) =  ∂ ∂ x φ ( x , y )    ∂ 2 ∂ y 2 φ ( x , y )  φ ( x , y ) − 1 +  ∂ ∂ y φ ( x , y )  2  φ ( x , y )  − 1 +  ∂ ∂ y φ ( x , y )  2 +  ∂ ∂ x φ ( x , y )  2  , 3 a 2 ( x, y ) =  ∂ ∂ y φ ( x, y )  ∂ 2 ∂ y 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + 2  ∂ ∂ x φ ( x, y )  ∂ 2 ∂ x∂ y φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + + − 3  ∂ ∂ y φ ( x, y )  3 − 2  ∂ ∂ x φ ( x, y )  2 ∂ ∂ y φ ( x, y ) + 3 ∂ ∂ y φ ( x, y ) φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , 3 a 3 ( x, y ) =  ∂ ∂ x φ ( x, y )  ∂ 2 ∂ x 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + 2  ∂ ∂ y φ ( x, y )  ∂ 2 ∂ x∂ y φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + + − 2  ∂ ∂ x φ ( x, y )   ∂ ∂ y φ ( x, y )  2 + 3 ∂ ∂ x φ ( x, y ) − 3  ∂ ∂ x φ ( x, y )  3 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , a 4 ( x, y ) =  ∂ ∂ y φ ( x, y )    ∂ ∂ x φ ( x, y )  2 +  ∂ 2 ∂ x 2 φ ( x, y )  φ ( x, y ) − 1  φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  . 4 The four-d imensional Riemann extension The metric (1) has a following co efficien ts o f connection Γ 1 11 =  ∂ ∂ x φ ( x, y )   1 −  ∂ ∂ x φ ( x, y )  2 +  ∂ 2 ∂ x 2 φ ( x, y )  φ ( x, y ) − 2  ∂ ∂ y φ ( x, y )  2  φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , Γ 2 11 =  ∂ ∂ y φ ( x, y )    ∂ ∂ x φ ( x, y )  2 +  ∂ 2 ∂ x 2 φ ( x, y )  φ ( x, y ) − 1  φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , Γ 1 12 = ∂ ∂ y φ ( x, y ) +  ∂ ∂ x φ ( x, y )   ∂ 2 ∂ x∂ y φ ( x, y )  φ ( x, y ) −  ∂ ∂ y φ ( x, y )  3 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , 9 Γ 2 12 = ∂ ∂ x φ ( x, y ) +  ∂ 2 ∂ x∂ y φ ( x, y )   ∂ ∂ y φ ( x, y )  φ ( x, y ) −  ∂ ∂ x φ ( x, y )  3 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , Γ 1 22 =  ∂ ∂ x φ ( x, y )    ∂ 2 ∂ y 2 φ ( x, y )  φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2  φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , Γ 1 22 =  ∂ ∂ y φ ( x, y )   − 2  ∂ ∂ x φ ( x, y )  2 + 1 −  ∂ ∂ y φ ( x, y )  2 +  ∂ 2 ∂ y 2 φ ( x, y )  φ ( x, y )  φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  . Using g iv en expressions for the connections co efficien ts w e in tro duce a 4 -dimensional Riemann space with the metric 4 ds 2 =  − 2Γ 1 ij z − 2Γ 2 ij t  dx i dx j + 2 dxdz + 2 dy d t (13) where z and t are an additional co ordinates. The Riemann space constructed o n suc h a w a y is called the Riemann extension of the base space equipp ed with connection [4 ]. In explicit fo rm the non zero comp onen ts of the metric (13 ) lo oks as g xx = − 2  ∂ ∂ x φ ( x, y )  z φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  +2  ∂ ∂ x φ ( x, y )  3 z φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2  ∂ ∂ x φ ( x, y )  z ∂ 2 ∂ x 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + 4  ∂ ∂ x φ ( x, y )  z  ∂ ∂ y φ ( x, y )  2 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2  ∂ ∂ y φ ( x, y )  t  ∂ ∂ x φ ( x, y )  2 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − 2  ∂ ∂ y φ ( x, y )  t ∂ 2 ∂ x 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + +2  ∂ ∂ y φ ( x, y )  t φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , g xy = − 2 z ∂ ∂ y φ ( x, y ) φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − 2  ∂ ∂ x φ ( x, y )  z ∂ 2 ∂ x∂ y φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + +2 z  ∂ ∂ y φ ( x, y )  3 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − 2 t ∂ ∂ x φ ( x, y ) φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2 t  ∂ 2 ∂ x∂ y φ ( x, y )  ∂ ∂ y φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + 2 t  ∂ ∂ x φ ( x, y )  3 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  , 10 g y y = − 2  ∂ ∂ x φ ( x, y )  z ∂ 2 ∂ y 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 + 2  ∂ ∂ x φ ( x, y )  z φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2  ∂ ∂ x φ ( x, y )  z  ∂ ∂ y φ ( x, y )  2 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  + 4  ∂ ∂ y φ ( x, y )  t  ∂ ∂ x φ ( x, y )  2 φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2  ∂ ∂ y φ ( x, y )  t φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  + 2  ∂ ∂ y φ ( x, y )  3 t φ ( x, y )  − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2  − − 2  ∂ ∂ y φ ( x, y )  t ∂ 2 ∂ y 2 φ ( x, y ) − 1 +  ∂ ∂ y φ ( x, y )  2 +  ∂ ∂ x φ ( x, y )  2 , g y t = 1 , g xz = 1 . Prop osition 1 Riemann sp ac e with the metric (13 ) is a Ric ci-flat R ij = 0 at the c o ndition − 2 ∂ ∂ x φ ( x, y ) ! 2 ∂ ∂ y φ ( x, y ) ! 2 − ∂ ∂ x φ ( x, y ) ! 4 + ∂ ∂ y φ ( x, y ) ! 2 + ∂ ∂ x φ ( x, y ) ! 2 − − 2 ∂ ∂ x φ ( x, y ) ! φ ( x, y ) ∂ ∂ y φ ( x, y ) ! ∂ 2 ∂ x∂ y φ ( x, y ) + ∂ 2 ∂ x 2 φ ( x, y ) ! φ ( x, y ) ∂ ∂ y φ ( x, y ) ! 2 + + ∂ 2 ∂ y 2 φ ( x, y ) ! ∂ 2 ∂ x 2 φ ( x, y ) ! ( φ ( x, y )) 2 + ∂ 2 ∂ y 2 φ ( x, y ) ! ∂ ∂ x φ ( x, y ) ! 2 φ ( x, y ) − − ∂ 2 ∂ y 2 φ ( x, y ) ! φ ( x, y ) − ∂ 2 ∂ x 2 φ ( x, y ) ! φ ( x, y ) − − ∂ 2 ∂ x∂ y φ ( x, y ) ! 2 ( φ ( x, y )) 2 − ∂ ∂ y φ ( x, y ) ! 4 = 0 . (14) Remark that t w o dimensional metric (1) at t his condition is a flat. It is imp or t an t to note that the space with the metric (13) with condition (1 4 ) do es not a flat, the comp onen t R 1212 of its Riemann tensor R 1212 6 = 0. So in result of the Riemann extension of t he metric (1) w e hav e got the Einstein space. Finally w e demonstrate some additional solutions of the equation (14 ). The substitution φ ( x, y ) = H ( x + y ) in to the equation (14) lead to t he condition on the function H ( x + y ) = H ( z ) 2 (D( H )( z )) 4 +  D (2)  ( H )( z ) H ( z ) − (D( H )( z )) 2 = 0 . 11 F rom here w e find the function H ( z ) in non explicit form q 2 ( H ( z )) 2 − C1 + C1 ln( − 2 C1 + 2 √ − C1 q 2 ( H ( z )) 2 − C1 H ( z ) ) 1 √ − C1 − z − C2 = 0 . The substitution φ ( x, y ) = H ( y x ) x lead to the complex solutions. In additiv e to the part (2 . 2) w e show the solutions of the the equation (14) ( or) whic h is connected with the function ω ( x, t ) in form ω ( x, t ) = A ( t ) + x √ t 2 − 1 , where A ( t ) is arbitrary function. In particular case A ( t ) = t 2 from here is follow ed that the function φ ( x, y ) defined from the equation − ( φ ( x, y )) 6 +  3 x 2 + 1 + 10 y + y 2  ( φ ( x, y )) 4 + +  − 2 y x 2 − 2 x 2 y 2 − 20 x 2 − 8 y − 8 y 3 − 3 x 4 − 32 y 2  ( φ ( x, y )) 2 + + y 2 x 4 − 8 x 4 y + 16 y 4 + x 6 + 16 y 2 + 32 y 3 + 8 x 2 y 2 + 32 y x 2 + 16 x 2 − 8 x 4 − 8 y 3 x 2 = 0 is the solution of the equation ( 1 4). References [1] V. Kagan, Osnovy te ori i p ov e rhnostei , v.2, OGIZ, Moskv a, (1948 ). [2] V. Dryuma, O n solutions of the he avenly e q uations and their gener alization s , ArXiv:gr- qc/0611001 v1, 31 Oct 20 0 6, pp.1-14. [3] V. D ryuma, On dual e quation in the ory of the se c ond or der ODE’s , ArXiv:nlin/0001 0 47 v1 22 Jan 20 07, pp.1-17. [4] V. Dryuma, The Riemann and Einsten-Weyl ge om e tries in the ory of differ en tial e quations, their applic ations and al l that . A.B.Shabat et all.(eds.), New T rends in Integrabilit y and P artial Solv a bilit y , Kluw er Academic Publishers, Printed in the Netherlands , 2004, p.11 5– 156. 12