On geometry of gonometric family of cycles
aa r X i v : . [ n li n . S I] O c t On geometry of gonometric family of cycles
Valerii Dryuma ∗ Institute of Mathematics and Informatics, AS RM,5 Academiei Street, 2028 Kishinev, Moldova , e-mail: [email protected]; [email protected] Abstract
An examples solutions of the equation for curvature of congruence of cycles areconsidered. Their properties are discussed.
Two parametrical family of cycles on the plane is determined by the equation( ξ − x ) + ( η − y ) − φ ( x, y ) = 0 . The angle metric in a given family of cycles is defined by the expression [1] ds = (cid:18) − (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) dx − ∂∂x φ ( x, y ) ∂∂y φ ( x, y ) dx dy + (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:19) dy ( φ ( x, y )) . (1)The expression for the curvature of the metric (1) has the form K ( x, y ) = φ (cid:18)(cid:18) − (cid:16) ∂∂y φ (cid:17) (cid:19) ∂ ∂x φ + 2 (cid:16) ∂∂y φ (cid:17) (cid:16) ∂∂x φ (cid:17) ∂ ∂x∂y φ + (cid:18) − (cid:16) ∂∂x φ (cid:17) (cid:19) ∂ ∂y φ (cid:19)(cid:18) − (cid:16) ∂∂x φ (cid:17) − (cid:16) ∂∂y φ (cid:17) (cid:19) −− φ (cid:18)(cid:16) ∂ ∂x φ (cid:17) ∂ ∂y φ − (cid:16) ∂ ∂x∂y φ (cid:17) (cid:19)(cid:18) − (cid:16) ∂∂x φ (cid:17) − (cid:16) ∂∂y φ (cid:17) (cid:19) − − ∂∂x φ ! − ∂∂y φ ! − + 1 . (2) ∗ Work supported in part by Grant RFFI, Russia-Moldova The congruence of cycles of constant curvature
The congruence of cycles of constant curvature K ( x, y ) = K are defined by the Monge-Ampereequation [1] K = φ (cid:18)(cid:18) − (cid:16) ∂∂y φ (cid:17) (cid:19) ∂ ∂x φ + 2 (cid:16) ∂∂y φ (cid:17) (cid:16) ∂∂x φ (cid:17) ∂ ∂x∂y φ + (cid:18) − (cid:16) ∂∂x φ (cid:17) (cid:19) ∂ ∂y φ (cid:19)(cid:18) − (cid:16) ∂∂x φ (cid:17) − (cid:16) ∂∂y φ (cid:17) (cid:19) −− φ (cid:18)(cid:16) ∂ ∂x φ (cid:17) ∂ ∂y φ − (cid:16) ∂ ∂x∂y φ (cid:17) (cid:19)(cid:18) − (cid:16) ∂∂x φ (cid:17) − (cid:16) ∂∂y φ (cid:17) (cid:19) − − ∂∂x φ ! − ∂∂y φ ! − + 1 . (3) For solutions of the Monge-Ampere equation F ( x, y, f x , f y , f xx , f xy , f yy ) = 0we use the method of solution of the p.d.e.’s described first in [2] and developed later in [3] .This method allow us to construct particular solutions of the partial nonlinear differentialequation F ( x, y, z, f x , f y , f z , f xx , f xy , f xz , f yy , f yz , f xxx , f xyy , f xxy , .. ) = 0 (4)with the help of transformation of the function and corresponding variables.Essence of the method consists in a following presentation of the functions and variables 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 yy → ( u t v t ) t v t , f xy → ( u x − u t v t v x ) t v t , ... (5)where variable t is considered as parameter.Remark that conditions of the type f xy = f yx , f xz = f zx ... are fulfilled at the such type of presentation.In result instead of equation (4) one get the relation between the new variables u ( x, t, z ), v ( x, t, z )and their partial derivatives Ψ( u, v, u x , u z , u t , v x , v z , v t ... ) = 0 . (6)This relation coincides with initial p.d.e at the condition v ( x, t, z ) = t and lead to the newp.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 expressedthrough the auxiliary function ω ( x, t, s ).Remark that there are a various means to reduce the relation (6) into the partial differentialequation.In a some cases the solution of equation (7) is a more simple problem than solution of equation(4). 2 emark 1 As the example we consider the Monge-Ampere equation ∂ ∂x f ( x, y ) ! ∂ ∂y f ( x, y ) − ∂ ∂x∂y f ( x, y ) ! + 1 = 0 . After the ( u, v ) -transformation v ( x, t ) = ∂∂t ω ( x, t ) ! t − ω ( x, t ) ,u ( x, t ) = ∂∂t ω ( x, t ) ! it takes the form of linear equation t ∂ ∂t ω ( x, t ) − ∂ ∂x ω ( x, t ) = 0 . with general solution ω ( x, t ) = t (cid:18) F1 ( − tx − t ) + F2 ( tx + 1 t ) (cid:19) , depending from two arbitrary functions.Choice of the functions Fi and elimination of the parameter t from corresponding relationslead to the function f ( x, y ) satisfying the Monge-Ampere equation. In the case K = 0 we get the equation φ ( x, y ) ∂ ∂x φ ( x, y ) − φ ( x, y ) ∂ ∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! +2 φ ( x, y ) ∂∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! ∂ ∂x∂y φ ( x, y ) + φ ( x, y ) ∂ ∂y φ ( x, y ) −− φ ( x, y ) ∂ ∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! − ( φ ( x, y )) ∂ ∂x φ ( x, y ) ! ∂ ∂y φ ( x, y )++ ( φ ( x, y )) ∂ ∂x∂y φ ( x, y ) ! − ∂∂x φ ( x, y ) ! − ∂∂y φ ( x, y ) ! ++ ∂∂x φ ( x, y ) ! + 2 ∂∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! + ∂∂y φ ( x, y ) ! = 0 (8)After applying of the (u,v)-transformation at this equation is reduced to the relation ∂∂t u ( x, t ) ! − ∂∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! −− ∂∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t )+36 ∂∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! −− u ( x, t ) ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂ ∂x u ( x, t ) + u ( x, t ) ∂∂t v ( x, t ) ! ∂ ∂t u ( x, t ) ! ∂∂x v ( x, t ) ! −− ( u ( x, t )) ∂∂t u ( x, t ) ! ∂ ∂x v ( x, t ) ! ∂ ∂t v ( x, t ) + 2 ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! −− ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! − u ( x, t )) ∂ ∂t∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂ ∂t∂x v ( x, t )++ ( u ( x, t )) ∂∂t u ( x, t ) ! ∂ ∂x v ( x, t ) ! ∂∂t v ( x, t ) ! ∂ ∂t u ( x, t )++ ( u ( x, t )) ∂ ∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂ ∂t v ( x, t )+( u ( x, t )) ∂∂t u ( x, t ) ! ∂ ∂t∂x v ( x, t ) ! −− u ( x, t ) ∂∂t v ( x, t ) ! ∂ ∂t u ( x, t ) ! ∂∂x u ( x, t ) ! − u ( x, t ) ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂ ∂x v ( x, t ) −− u ( x, t ) ∂∂t v ( x, t ) ! ∂ ∂t∂x u ( x, t ) ! ∂∂x v ( x, t ) − ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! ∂∂x v ( x, t ) −− u ( x, t ) ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂ ∂t v ( x, t ) − u ( x, t ) ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! ∂ ∂t v ( x, t )++2 u ( x, t ) ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! ∂ ∂t∂x u ( x, t )++2 u ( x, t ) ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! ∂ ∂t∂x v ( x, t ) −− u ( x, t ) ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! ∂ ∂t∂x v ( x, t )++ u ( x, t ) ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂ ∂t v ( x, t ) ! ∂∂x u ( x, t ) ! −− ( u ( x, t )) ∂ ∂x u ( x, t ) ! ∂∂t v ( x, t ) ! ∂ ∂t u ( x, t )+2 ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t )++ u ( x, t ) ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂ ∂x v ( x, t ) + ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! ++ ∂∂x u ( x, t ) ! ∂∂t v ( x, t ) ! − ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! −− ∂∂t v ( x, t ) ! ∂∂t u ( x, t ) ! ∂∂x v ( x, t ) ! + 2 ∂∂t u ( x, t ) ! ∂∂t v ( x, t ) ! ∂∂x u ( x, t ) ! +4 u ( x, t ) ∂∂t v ( x, t ) ! ∂ ∂t u ( x, t ) + u ( x, t ) ∂∂t v ( x, t ) ! ∂ ∂x u ( x, t )++ ( u ( x, t )) ∂ ∂t∂x u ( x, t ) ! ∂∂t v ( x, t ) ! = 0 . From here after the choice of the functions u and v in the form v ( x, t ) = t ∂∂t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂∂t ω ( x, t )we find the equation − ∂ ∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! − t ∂∂t ω ( x, t ) − ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t )++ ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t ) + ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! −− ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! − ∂∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! ++ ∂ ∂t ω ( x, t ) − ∂ ∂t ω ( x, t ) ! t + ∂ ∂t ω ( x, t ) ! ∂∂x ω ( x, t ) ! ++2 ∂∂x ω ( x, t ) ! ∂ ∂t ω ( x, t ) + 2 ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! ∂∂x ω ( x, t )++ ∂∂t ω ( x, t ) ! ∂ ∂x ω ( x, t ) = 0having the particular solution ω ( x, t ) = A ( t ) + x, where − d dt A ( t ) ! t − t ddt A ( t ) + 4 d dt A ( t ) = 0 . General solution of this equation is A ( t ) = C1 + C2 ln( t + √ t − t from the relations y √ t − − t C2 + C1 √ t − C2 ln( t + √ t − √ t − x √ t − , and φ ( x, y ) √ t − − C2 = 0give us the function φ ( x, y ) defined from the equation y − q φ ( x, y )) + 1 + C1 + ln( q φ ( x, y )) + 1 + 1 φ ( x, y ) ) + x = 0satisfying the equation (8). 5 .3 Congruence of positive constant curvature In the case K = 1 from (3) we get the equation φ ( x, y ) ∂ ∂x φ ( x, y ) − φ ( x, y ) ∂ ∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! ++2 φ ( x, y ) ∂∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! ∂ ∂x∂y φ ( x, y )++ φ ( x, y ) ∂ ∂y φ ( x, y ) − φ ( x, y ) ∂ ∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! −− ( φ ( x, y )) ∂ ∂x φ ( x, y ) ! ∂ ∂y φ ( x, y ) + ( φ ( x, y )) ∂ ∂x∂y φ ( x, y ) ! − ∂∂x φ ( x, y ) ! + ∂∂y φ ( x, y ) ! = 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 ) ! ∂ ∂x ω ( x, t ) + ∂ ∂t ω ( x, t ) ! t − ∂ ∂t ω ( x, t ) ! t ++2 ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! ∂∂x ω ( x, t ) + ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t ) −− ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t ) − ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! ++ ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! + ∂ ∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! − t ∂∂t ω ( x, t ) −− ∂∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! = 0 . (10)This equation admits particular solution in the form ω ( x, t ) = A ( t ) + x where the function A ( t ) is solution of equation2 d dt A ( t ) ! t − d dt A ( t ) ! t − t ddt A ( t ) = 0 . So we get A ( t ) = C1 + C2 √− t t from the relations y √− t + C1 √− t − C2 + x √− t = 0and φ ( x, y ) √ t − − C2 = 0we obtain the simplest solution of the equation (9). φ ( x, y ) = 1 / q y + 4 y C1 + 4 yx + 2 C1 + 4 C1 x + 2 x + 4 C2 . Moore complicated solutions of the equation (10) in the form ω ( x, t ) = A ( t ) + B ( t ) x lead to the conditions B ( t ) = −√ t − , and A ( t )is arbitrary function.In result elimination of the parameter t from the relations y √ t − − t ddt A ( t ) ! √ t − A ( t ) √ t − x = 0 , and φ ( x, y ) √ t − − ddt A ( t ) ! √ t − tx = 0with a given function A ( t ) we get the solution of the equation (9) dependent from choice ofarbitrary function.As example in the case A ( t ) = 1 t we find the solution of the equation (9) in the form16 ( φ ( x, y )) + (cid:16) y − x − (cid:17) ( φ ( x, y )) + (cid:16) − y + 16 − x + x + y + 2 y x (cid:17) ( φ ( x, y )) ++ (cid:16) y − y + 8 x − y x + 32 x (cid:17) φ ( x, y ) − y − x +20 y x − x − y x − x − y x + y = 0 . In the case K = − φ ( x, y ) ∂ ∂x φ ( x, y ) − φ ( x, y ) ∂ ∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! ++2 φ ( x, y ) ∂∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! ∂ ∂x∂y φ ( x, y ) + φ ( x, y ) ∂ ∂y φ ( x, y ) − φ ( x, y ) ∂ ∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! − ( φ ( x, y )) ∂ ∂x φ ( x, y ) ! ∂ ∂y φ ( x, y )++ ( φ ( x, y )) ∂ ∂x∂y φ ( x, y ) ! + 1 − ∂∂x φ ( x, y ) ! − ∂∂y φ ( x, y ) ! ++2 ∂∂x φ ( x, y ) ! + 4 ∂∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! + 2 ∂∂y φ ( x, y ) ! = 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 ) ! ∂ ∂x ω ( x, t ) + ∂ ∂t ω ( x, t ) ! t − ∂ ∂t ω ( x, t ) ! t ++2 ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! ∂∂x ω ( x, t ) + ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t ) −− ∂ ∂t ω ( x, t ) ! ∂∂t ω ( x, t ) ! t ∂ ∂x ω ( x, t ) − ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! ++ ∂∂t ω ( x, t ) ! t ∂ ∂t∂x ω ( x, t ) ! + ∂ ∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! − t ∂∂t ω ( x, t ) −− ∂∂t ω ( x, t ) ! t ∂∂x ω ( x, t ) ! = 0 . (12)This equation admits the particular solution ω ( x, t ) = √− t + 1 x + A ( t )with arbitrary function A ( t ).In particular case after elimination of the parameter t from the relations y √− t + 1 + xt − t √− t + 1 = 0and φ ( x, y ) √− t + 1 − t √− t + 1 + x = 0we find the solution of the equation (11) in the form −
16 ( φ ( x, y )) + (cid:16) −
32 + 8 y − x (cid:17) ( φ ( x, y )) ++ (cid:16) − y x + 32 y − − y + 8 x − x (cid:17) ( φ ( x, y )) ++ (cid:16) − y x − y + 32 x + 8 x + 8 y (cid:17) φ ( x, y ) −− y + 16 x + 3 x y + 8 x + 3 y x + x + y − y x = 08 Geodesic equations
The geodesic of the metric (1) are equivalent to the equation d dx y ( x ) + a 1 ( x, y ) ddx y ( x ) ! + 3 a 2 ( x, y ) ddx y ( x ) ! + 3 a 3 ( x, y ) ddx y ( x ) + a 4 ( x, y ) = 0 , where a 1 ( x , y ) = (cid:16) ∂∂ x φ ( x , y ) (cid:17) (cid:18)(cid:16) ∂ ∂ y φ ( x , y ) (cid:17) φ ( x , y ) − + (cid:16) ∂∂ y φ ( x , y ) (cid:17) (cid:19) φ ( x , y ) (cid:18) − + (cid:16) ∂∂ y φ ( x , y ) (cid:17) + (cid:16) ∂∂ x φ ( x , y ) (cid:17) (cid:19) , a 2 ( x, y ) = (cid:16) ∂∂y φ ( x, y ) (cid:17) ∂ ∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) + 2 (cid:16) ∂∂x φ ( x, y ) (cid:17) ∂ ∂x∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) ++ − (cid:16) ∂∂y φ ( x, y ) (cid:17) − (cid:16) ∂∂x φ ( x, y ) (cid:17) ∂∂y φ ( x, y ) + 3 ∂∂y φ ( x, y ) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , a 3 ( x, y ) = (cid:16) ∂∂x φ ( x, y ) (cid:17) ∂ ∂x φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) + 2 (cid:16) ∂∂y φ ( x, y ) (cid:17) ∂ ∂x∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) ++ − (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:16) ∂∂y φ ( x, y ) (cid:17) + 3 ∂∂x φ ( x, y ) − (cid:16) ∂∂x φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , a 4 ( x, y ) = (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:18)(cid:16) ∂∂x φ ( x, y ) (cid:17) + (cid:16) ∂ ∂x φ ( x, y ) (cid:17) φ ( x, y ) − (cid:19) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) . The metric (1) has a following coefficients of connectionΓ = (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:18) − (cid:16) ∂∂x φ ( x, y ) (cid:17) + (cid:16) ∂ ∂x φ ( x, y ) (cid:17) φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:19) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , Γ = (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:18)(cid:16) ∂∂x φ ( x, y ) (cid:17) + (cid:16) ∂ ∂x φ ( x, y ) (cid:17) φ ( x, y ) − (cid:19) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , Γ = ∂∂y φ ( x, y ) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:16) ∂ ∂x∂y φ ( x, y ) (cid:17) φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , = ∂∂x φ ( x, y ) + (cid:16) ∂ ∂x∂y φ ( x, y ) (cid:17) (cid:16) ∂∂y φ ( x, y ) (cid:17) φ ( x, y ) − (cid:16) ∂∂x φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , Γ = (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:18)(cid:16) ∂ ∂y φ ( x, y ) (cid:17) φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:19) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , Γ = (cid:16) ∂∂y φ ( x, y ) (cid:17) (cid:18) − (cid:16) ∂∂x φ ( x, y ) (cid:17) + 1 − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂ ∂y φ ( x, y ) (cid:17) φ ( x, y ) (cid:19) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) . Using given expressions for the connections coefficients we introduce a 4-dimensional Riemannspace with the metric ds = (cid:16) − ij z − ij t (cid:17) dx i dx j + 2 dxdz + 2 dydt (13)where z and t are an additional coordinates.The Riemann space constructed on such a way is called the Riemann extension of the basespace equipped with connection [4].In explicit form the non zero components of the metric (13) looks as g xx = − (cid:16) ∂∂x φ ( x, y ) (cid:17) zφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) +2 (cid:16) ∂∂x φ ( x, y ) (cid:17) zφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− (cid:16) ∂∂x φ ( x, y ) (cid:17) z ∂ ∂x φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) + 4 (cid:16) ∂∂x φ ( x, y ) (cid:17) z (cid:16) ∂∂y φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− (cid:16) ∂∂y φ ( x, y ) (cid:17) t (cid:16) ∂∂x φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) − (cid:16) ∂∂y φ ( x, y ) (cid:17) t ∂ ∂x φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) ++2 (cid:16) ∂∂y φ ( x, y ) (cid:17) tφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) ,g xy = − z ∂∂y φ ( x, y ) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) − (cid:16) ∂∂x φ ( x, y ) (cid:17) z ∂ ∂x∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) ++2 z (cid:16) ∂∂y φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) − t ∂∂x φ ( x, y ) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− t (cid:16) ∂ ∂x∂y φ ( x, y ) (cid:17) ∂∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) + 2 t (cid:16) ∂∂x φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) , yy = − (cid:16) ∂∂x φ ( x, y ) (cid:17) z ∂ ∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) + 2 (cid:16) ∂∂x φ ( x, y ) (cid:17) zφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− (cid:16) ∂∂x φ ( x, y ) (cid:17) z (cid:16) ∂∂y φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) + 4 (cid:16) ∂∂y φ ( x, y ) (cid:17) t (cid:16) ∂∂x φ ( x, y ) (cid:17) φ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− (cid:16) ∂∂y φ ( x, y ) (cid:17) tφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) + 2 (cid:16) ∂∂y φ ( x, y ) (cid:17) tφ ( x, y ) (cid:18) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) (cid:19) −− (cid:16) ∂∂y φ ( x, y ) (cid:17) t ∂ ∂y φ ( x, y ) − (cid:16) ∂∂y φ ( x, y ) (cid:17) + (cid:16) ∂∂x φ ( x, y ) (cid:17) ,g yt = 1 , g xz = 1 . Proposition 1
Riemann space with the metric (13) is a Ricci-flat R ij = 0 at the condition − ∂∂x φ ( x, y ) ! ∂∂y φ ( x, y ) ! − ∂∂x φ ( x, y ) ! + ∂∂y φ ( x, y ) ! + ∂∂x φ ( x, y ) ! −− ∂∂x φ ( x, y ) ! φ ( x, y ) ∂∂y φ ( x, y ) ! ∂ ∂x∂y φ ( x, y ) + ∂ ∂x φ ( x, y ) ! φ ( x, y ) ∂∂y φ ( x, y ) ! ++ ∂ ∂y φ ( x, y ) ! ∂ ∂x φ ( x, y ) ! ( φ ( x, y )) + ∂ ∂y φ ( x, y ) ! ∂∂x φ ( x, y ) ! φ ( x, y ) −− ∂ ∂y φ ( x, y ) ! φ ( x, y ) − ∂ ∂x φ ( x, y ) ! φ ( x, y ) −− ∂ ∂x∂y φ ( x, y ) ! ( φ ( x, y )) − ∂∂y φ ( x, y ) ! = 0 . (14)Remark that two dimensional metric (1) at this condition is a flat.It is important to note that the space with the metric (13) with condition (14) does not a flat,the component R of its Riemann tensor R = 0.So in result of the Riemann extension of the metric (1) we have got the Einstein space.Finally we demonstrate some additional solutions of the equation (14).The substitution φ ( x, y ) = H ( x + y )into the equation (14) lead to the condition on the function H ( x + y ) = H ( z )2 (D( H )( z )) + (cid:16) D (2) (cid:17) ( H )( z ) H ( z ) − (D( H )( z )) = 0 . H ( z ) in non explicit form q H ( z )) − C1 + C1 ln( − C1 + 2 √− C1 q H ( z )) − C1 H ( z ) ) 1 √− C1 − z − C2 = 0 . The substitution φ ( x, y ) = H ( yx ) x lead to the complex solutions.In additive to the part (2 .
2) we show the solutions of the the equation (14) (or) which isconnected with the function ω ( x, t ) in form ω ( x, t ) = A ( t ) + x √ t − , where A ( t ) is arbitrary function.In particular case A ( t ) = t from here is followed that the function φ ( x, y ) defined from theequation − ( φ ( x, y )) + (cid:16) x + 1 + 10 y + y (cid:17) ( φ ( x, y )) ++ (cid:16) − yx − x y − x − y − y − x − y (cid:17) ( φ ( x, y )) ++ y x − x y + 16 y + x + 16 y + 32 y + 8 x y + 32 yx + 16 x − x − y x = 0is the solution of the equation (14). References [1] V. Kagan,
Osnovy teorii poverhnostei , v.2, OGIZ, Moskva, (1948).[2] V. Dryuma,
On solutions of the heavenly equations and their generalizations , ArXiv:gr-qc/0611001 v1, 31 Oct 2006, pp.1-14.[3] V. Dryuma,
On dual equation in theory of the second order ODE’s , ArXiv:nlin/0001047 v122 Jan 2007, pp.1-17.[4] V. Dryuma,