aa r X i v : . [ phy s i c s . g e n - ph ] N ov On equations defining the Ricci flows of manifolds
Valerii Dryuma ∗ Institute of Mathematics and Informatics, AS RM,5 Academiei Street, 2028 Kishinev, Moldova , e-mail: [email protected]; [email protected] Abstract
The examples of the Ricci flows of the four-dimensional manifolds which are determinedby help of nonlinear differential equations of the type of Monge-Ampere are constructed.Theirparticular solutions are derived and their properties are discussed. D model of the Ricci-flow We study the system of equations ∂∂t g ij ( ~x, t ) = − R ij ( g ) (1)describing the Ricci flows of the four dimensional manifolds which are endowed by the metrics ofthe form ds = A ( x, y, t ) du + 2 B ( x, y, t ) du dv + du dx + C ( x, y, t ) dv + dv dy , (2)where the components of metrics are dependent from two coordinates ( x, y ) and from the param-eter t .In this case the Ricci-tensor of the metric (1) has a five components R uu , R uv , R vv , R ux , R vx . From the conditions of compatibility of the equations (1) we find that they are reduced to theequation 4 ∂ ∂x h ( x, y, t ) ! ∂ ∂y h ( x, y, t ) − ∂ ∂x∂y h ( x, y, t ) ! − ∂∂t h ( x, y, t ) = 0 (3)and components of the metric (2) take the form B ( x, y, t ) = ∂ ∂x∂y h ( x, y, t ) , C ( x, y, t ) = − ∂ ∂x h ( x, y, t ) , A ( x, y, t ) = − ∂ ∂y h ( x, y, t ) . (4) ∗ Work supported in part by Grant RFFI, Russia-Moldova τ and has form of the Monge-Ampere equation relatively of the variables x, y Let us consider some elementary solutions of equation (3)After the substitution of the form h ( x, y, t ) = H xy − t ! y the equation (3) takes the form48 (cid:16) D (2) (cid:17) ( H )( η ) H ( η ) −
36 (D( H )( η )) + D( H )( η ) = 0 , (5)where η = xy − t. Solution of the equation (4) may be present in form η ( H ) = C2 − √ HK + −
48 ln(1 − √ HK ) + 24 ln(1 + √ HK + √ HK ) + 48 √ √ √ HK √ HK ) ! K − , where K and C2 are constants.Remark that if the value K = − η ( H ) has a break.The simplest singular solution of (4) is h ( x, y, t ) == (cid:16) B x k + C1 √ x cos(1 / √
23 ln( x )) + C2 √ x sin(1 / √
23 ln( x )) + yBx + 1 / y kx (cid:17) (2 kt + C4 ) − . More complicated solution has the form h ( x, y, t ) = − / C1 µ √ C1 c √ / √ C1 c ( x − t ) √ c + µ C1 y c −− / µ √ C1 c √ y c tan(1 / √ C1 c ( x − t ) √ C1 . A more general solutions of the equation (4) are derived by the method of parametric repre-sentations of functions and their derivatives [1-2].Let us apply its to the equation (4).After the change of the variables, the function and its derivative h ( x, y, t ) → u ( x, τ, t ) , y → v ( x, τ, t ) , h x → u x − u τ v τ v x = p, h t → u t − u τ v τ v t = q, y → u τ v τ = r, h xx → p x − p τ v τ v x , h xy → p τ v τ = q x − q τ v τ v x , h yy → r τ v τ (6)we derive from the equation ∂ ∂x h ( x, y, t ) ! ∂ ∂y h ( x, y, t ) − ∂ ∂x∂y h ( x, y, t ) ! − ∂∂t h ( x, y, t ) = 0 (7)the relation between the functions u ( x, τ, t ) and v ( x, τ, t ), where variable τ is considered as pa-rameter − ∂ ∂x u ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! ∂∂τ u ( x, τ, t ) ! ∂ ∂τ v ( x, τ, t ) −− ∂∂τ u ( x, τ, t ) ! ∂ ∂x v ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! ∂ ∂τ u ( x, τ, t )++ ∂∂τ u ( x, τ, t ) ! ∂ ∂x v ( x, τ, t ) ! ∂ ∂τ v ( x, τ, t ) −− ∂ ∂τ ∂x u ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! + ∂ ∂x u ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! ∂ ∂τ u ( x, τ, t )++2 ∂ ∂τ ∂x u ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! ∂∂τ u ( x, τ, t ) ! ∂ ∂τ ∂x v ( x, τ, t ) −− ∂∂τ u ( x, τ, t ) ! ∂ ∂τ ∂x v ( x, τ, t ) ! − ∂∂t u ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! ++ ∂∂τ u ( x, τ, t ) ! ∂∂t v ( x, τ, t ) ! ∂∂τ v ( x, τ, t ) ! = 0 . (8)Note that the relation (8) under the condition v ( x, τ, t ) = τ is equivalent to the equation ∂ ∂x u ( x, t, τ ) ∂ ∂t u ( x, t, τ ) − ∂ ∂t∂x u ( x, t, τ ) ! − ∂∂τ u ( x, t, τ ) = 0similar to equation (7) and transformed into an equation of more general form if the functions u ( x, τ, t ), ( v ( x, τ, t ) are dependent.For example under the condition u ( x, τ, t ) = τ ∂∂τ ω ( x, τ, t ) − ω ( x, τ, t ) , v ( x, τ, t ) = ∂∂τ ω ( x, τ, t )from the (8) we obtain the p.d.e. for the function ω ( x, τ, t ) − ∂ ∂x ω ( x, τ, t ) + ∂ ∂τ ω ( x, τ, t ) ! ∂∂t ω ( x, τ, t ) = 0 . (9)The other condition v ( x, τ, t ) = τ ∂∂τ ω ( x, τ, t ) − ω ( x, τ, t ) , u ( x, τ, t ) = ∂∂τ ω ( x, τ, t )3ead to the equation ∂ ∂x ω ( x, τ, t ) + ∂ ∂τ ω ( x, τ, t ) ! τ ∂∂t ω ( x, τ, t ) = 0 . (10)Let us consider some solutions of the equation (9).Under the substitution ω ( x, τ, t ) = A ( x − kt, τ )it is reduced at the equation ∂ ∂η A ( η, τ ) + k ∂ ∂τ A ( η, τ ) ! ∂∂η A ( η, τ ) = 0 (11)where η = x − kt .This equation is reduced after the Legendre transformation to the Euler-Trikomi equation andtherefore its solutions depend radically on the sign of the parameter k .Each solution (11) corresponds to the solution of the original equation (8) which is obtainedby eliminating the parameter τ from the correlations y − τ ∂∂τ ω ( x, τ, t ) + ω ( x, τ, t ) = 0 , h ( x, y, t ) − ∂∂τ ω ( x, τ, t ) = 0 . (12)Here is an example of. After the substitutions A ( x, y, t ) = A ( t ) + A ( t ) x + A ( t ) y + A ( t ) x + A ( t ) yx + A ( t ) y ,B ( x, y, t ) = B ( t ) + B ( t ) x + B ( t ) y + B ( t ) x + B ( t ) xy + B ( t ) y ,C ( x, y, t ) = C ( t ) + C ( t ) x + C ( t ) y + C ( t ) x + C ( t ) xy + C ( t ) y , B ( t ) = − A ( t ) , A ( t ) = − B ( t ) , C ( t ) = − B ( t ) , C ( t ) = A ( t )the metric (2) takes the form ds = (cid:16) A ( t ) + A ( t ) x + A ( t ) y + A ( t ) x − B ( t ) yx + A ( t ) y (cid:17) du ++2 (cid:16) B ( t ) + B ( t ) x + B ( t ) y + B ( t ) x − A ( t ) xy + B ( t ) y (cid:17) du dv ++ (cid:16) C ( t ) + C ( t ) x + C ( t ) y + C ( t ) x − B ( t ) xy + A ( t ) y (cid:17) dv + dx du + dy dv (13)and from the system of equations (1) is obtained system of ODE’s ddt B ( t ) = 24 B ( t ) A ( t ) − C ( t ) B ( t ) ,ddt C ( t ) = − C ( t ) A ( t ) + 48 ( B ( t )) , dt A ( t ) = − C ( t ) A ( t ) + 24 ( A ( t )) − B ( t ) B ( t ) ,ddt B ( t ) = 24 A ( t ) B ( t ) − B ( t ) A ( t ) ,ddt A ( t ) = − A ( t ) A ( t ) + 48 ( B ( t )) (14)on the functions A ( t ) , B ( t ) , C ( t ) , A ( t ) , B ( t ) from which expressed the rest coefficients ofthe metric (13).The system of equations (14) has a first integral M of the type C ( t ) B ( t ) + M C ( t ) B ( t ) − A ( t ) B ( t ) + 2 M B ( t ) − M C ( t ) A ( t ) B ( t ) = 0with the help of which the order can be reduced.In result we get the system with parameter ddt B ( t ) = 24 B ( t ) A ( t ) − C ( t ) B ( t ) , ddt C ( t ) = − C ( t ) A ( t ) + 48 ( B ( t )) , B ( t ) ddt A ( t ) = − C ( t ) B ( t ) − M C ( t ) B ( t ) − C ( t ) M B ( t ) ++24 M C ( t ) A ( t ) B ( t ) + 24 A ( t ) B ( t ) − B ( t ) B ( t ) , B ( t ) ddt B ( t ) = 24 A ( t ) B ( t ) B ( t ) − C ( t ) B ( t ) − M C ( t ) B ( t ) −− M B ( t ) + 72 M C ( t ) A ( t ) B ( t ) . Theorem 1
The flows of Ricci of the D - manifolds which are endowed by the metric with localcoordinates ( x, y, z, t ) the components of which are dependent from two coordinates ( x, t ) and fromthe parameter τ ds = ∂ ∂x f ( x, t, τ ) ! dx + 2 ∂ ∂t∂x f ( x, t, τ ) ! dx dt + ∂ ∂t f ( x, t, τ ) ! dy ++2 ∂ ∂t∂x f ( x, t, τ ) ! dy dz + ∂ ∂x f ( x, t, τ ) ! dz + ∂ ∂t f ( x, t, τ ) ! dt is defined by the equation ∂∂τ f ( x, t, τ ) = ln ∂ ∂t f ( x, t, τ ) ∂ ∂x f ( x, t, τ ) − ∂ ∂t∂x f ( x, t, τ ) ! (15)Let us consider some solutions of the equation (15).With aim of convenience we rewrite if in the form ∂ ∂x f ( x, y, τ ) ! ∂ ∂y f ( x, y, τ ) − ∂ ∂x∂y f ( x, y, τ ) ! − e ∂∂τ f ( x,y,τ ) = 0 , (16)5here the variable t is changed on y .After the ( u, v )-transformation with condition u ( x, t, τ ) = t the equation (16) takes the form ∂ ∂x v ( x, t, τ ) ! ∂ ∂t v ( x, t, τ ) − ∂ ∂t∂x v ( x, t, τ ) ! − e − ∂∂τ v ( x,t,τ ) ∂∂t v ( x,t,τ ) ∂∂t v ( x, t, τ ) ! = 0 . Particular solutions of this equation are obtained with the help of additional conditions. Forexample in the case ∂∂τ v ( x, t, τ ) = ∂∂t v ( x, t, τ )we find that the function v ( x, t, τ ) has the form v ( x, t, τ ) = F1 ( x, τ + t ) = h ( x, η ) , where the function h ( x, η ) satisfies the equation ∂ ∂η h ( x, η ) ! ∂ ∂x h ( x, η ) − ∂ ∂η∂x h ( x, η ) ! − e − ∂∂η h ( x, η ) ! = 0 . (17)After the ( u, v )-transformation with the conditions v ( x, t ) = t ∂∂t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂∂t ω ( x, t )the equation (17) is reduced to the linear equation ∂ ∂x ω ( x, t ) + ∂ ∂t ω ( x, t ) ! e − = 0with general solution ω ( x, t ) = F1 ( t − i √ e − x ) + F2 ( t + i √ e − x )containing two arbitrary functions.With the help of the function ω ( x, t ) we can obtain large class of solutions of the equation (16)in parametric form.For example, in the case F1 ( t − i √ e − x ) = cosh(+ − i √ e − x ) F2 ( t + i √ e − x ) = sinh( t + i √ e − x )we get ω ( x, t ) = cosh( t ) cos( √ e − x ) + sinh( t ) cos( √ e − x )and elimination of the parameter t from the relations η − t cos( e − / x ) cosh( t ) − t sinh( t ) cos( e − / x ) + cosh( t ) cos( e − / x ) + sinh( t ) cos( e − / x ) = 0 ,h ( x, η ) − cosh( t ) cos( e − / x ) − sinh( t ) cos( e − / x )lead to the function h ( x, η ) = e LambertW ( η e − e − / x ) )+1 cos( e − / x )which is solution of the equation (17).With the help of solutions of the equation (17) can be constructed solutions of the equation(16). 6 More general solution
The equation (16) after the ( u, v )-transformation with conditions − − (cid:16) ∂∂τ u ( x, t, τ ) (cid:17) ∂∂t v ( x, t, τ ) + (cid:16) ∂∂t u ( x, t, τ ) (cid:17) ∂∂τ v ( x, t, τ ) ∂∂t v ( x, t, τ ) = Ax,v ( x, t, τ ) = F1 ( x, u ( x, t, τ ) − Axτ takes the form ∂ ∂η h ( x, η ) ! ∂ ∂x h ( x, η ) − ∂ ∂η∂x h ( x, η ) ! − e Ax ∂∂η h ( x, η ) ! = 0 , (18)where η = u ( x, t, τ ) − Axτ,h ( x, η ) = F1 ( x, u ( x, t, τ ) − Axτ ) . It is reduced to the equation e Ax ∂ ∂t ω ( x, t ) + ∂ ∂x ω ( x, t ) = 0 (19)after the ( u, v )-transformation with conditions v ( x, t ) = t ∂∂t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂∂t ω ( x, t ) . Simplest solution of the equation (19) has the form ω ( x, t ) = ( C1 cos( √ c t ) + C2 sin( √ c t )) ·· (cid:18) C3 BesselJ (0 , r − c A √ e Ax ) + C4 BesselY (0 , r − c A √ e Ax ) (cid:19) and elimination of the parameter t from the relations η + t sin( √ c t ) √ c BesselJ (0 , √− c √ e x ) + cos( √ c t ) BesselJ (0 , √− c √ e x ) = 0 ,h ( x, η ) + sin( √ c t ) √ c BesselJ (0 , √− c √ e x ) = 0gives solution of the equation (18) defined from the relation η √ c + arcsin h ( x, η ) √ c BesselJ (0 , √− c √ e x ) ! h ( x, η )++ r c (cid:16) BesselJ (0 , √− c √ e x ) (cid:17) − ( h ( x, η )) = 0 , (at the conditions A = 1 and C2 = , C4 = , C1 = ).The solution of the equation (16) which corresponds the function h ( x, η ) is determined fromthe relation y − h ( x, f ( x, y, τ ) − xτ ) = 0and can be very complicated. References:
1. V. Dryuma,
On nonlinear equations associated with developable, ruled and minimal surfaces .ArXiv:1002.0952 v1 [physycs.gen-ph] 4 Feb.2010 p. 1-13.2. V. Dryuma,
The Riemann and Einsten-Weyl geometries in theory of differential equations, theirapplications and all that . A.B.Shabat et all.(eds.), New Trends in Integrability and Partial Solvability,Kluwer Academic Publishers, Printed in the Netherlands , 2004, p.115–156.. A.B.Shabat et all.(eds.), New Trends in Integrability and Partial Solvability,Kluwer Academic Publishers, Printed in the Netherlands , 2004, p.115–156.