Burgers ’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

We construct metric connection associated with a first-order differential equation bymeans of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt = u(t, x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.


Introduction
A simple dynamical transport phenomenon such as transport of a pollutant in a fluid, transportation of information, wave propagation, traffic flow, population density, chemical reactions, and gas dynamics is described by a certain first-order partial differential equation (PDE) of space and time variables, the so-called transport equation [1,2].Among them, Burgers' equation may be the more intriguing one which serves as a prototype model for turbulent flows [3].Many works in the literature are devoted to investigation of Lie symmetries and the exact solutions of Burgers' equation(s) and its generalizations [4][5][6][7][8][9][10][11][12][13][14].Beyond its importance in mathematical physics, it appears also as a geodesic equation on diffeomorphism group of a circle with right invariant  2 metric (see [15] and references therein).As we show in this paper, the identity for the Gaussian curvature of a surface associated with a first-order ordinary differential equation (ODE) in the first-order jet bundle is described by an inhomogeneous Burgers' equation: Geometric study of differential equations is a notable subject in differential geometry and dates back to Tresse, Lie, and Cartan [16][17][18][19].A considerable number of works in the literature are devoted to study of problems such as linearization, existence of local coordinates in which an equation takes the special form, and theory of connections associated with differential equations [20][21][22][23][24][25][26][27].In modern terminology, an ODE or a PDE is identified with an exterior differential system in an appropriate jet bundle and this approach leads to geometrization of differential equations.In this work, we basically deal with the first-order ODEs and PDEs in the context of Riemannian geometry and Riemannian structure defined on a submanifold corresponding to given equation enables one to give a geometric description of an interesting class of equations in mathematical physics.
With this regard, in this paper, we construct o(2, R) and o(3, R)-valued connections from the exterior differential systems encoding first-order ODEs and PDEs on corresponding submanifolds in the appropriate jet bundles.As a new result, we firstly show that the Gaussian curvature of a surface corresponding to a first-order ODE in  1 (R, R) is described by nonlinear PDE (1).Accordingly, we show that the inviscid and viscous Burgers' equations and Korteweg-de 2 Advances in Mathematical Physics Vries (KdV) equation can be seen as the equations for surfaces with certain Gaussian curvatures.Also, Riemannian metrics for spaces of constant curvature can be attached to some linear ODEs by solving certain inhomogeneous Burgers' equations.For a system of first-order PDEs, we show that an integral manifold of any system of first-order PDEs defines intrinsically flat and totally geodesic submanifold.We prove that the scalar curvature of three-dimensional Riemannian manifold obtained from a system of first-order PDEs is given by the sum of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the characteristic equations.Accordingly, we show that, for all points on an integral submanifold of a flat manifold, sectional curvatures are described by a pair of inhomogeneous Burgers' equations.

Basics of Riemannian Geometry in an Orthonormal Frame.
In this subsection, we shall give the basic notions of the formulation of the Riemannian geometry in an orthonormal frame.We assume that all considerations are local and all objects are smooth.For details in the subject we refer to [28][29][30].
Let (M, ) be a  dimensional Riemannian manifold and let  be a coordinate neighbourhood of M. Let Γ  (M) and Γ  ( * M) denote the set of sections of the tangent and cotangent bundles of M over , respectively.For simplicity, let us denote Γ  by Γ.Let   ∈ Γ(M) and   ∈ Γ( * M) for  = 1, . . ., .An orthonormal frame field ( 1 , . . .,   ) on , which is dual to the coframe field ( 1 , . . .,   ), is defined with respect to symmetric section  of  * M ⊗  * M as here   (  ) =    . is called the Riemannian metric on .The matrix of 1-forms  = (   ), satisfying Clearly one has    = −   .Exterior derivative of (3) gives 0 = (   +    ∧    ) ∧   .This implies that Ω = (Ω   ) is called curvature 2-form or the Riemannian curvature tensor associated with the connection  and it is written as Riemannian curvature tensor has the symmetries Ricci curvature and the scalar curvature are defined, respectively, by The covariant derivative operator associated with the connection  is defined by with

Jet Bundle Formulation of the First-Order Equations.
Consider a first-order ODE of the form For the function (, , ) =  − (, ), (11) can be described geometrically as a surface in the first-order jet bundle by  −1 (0).That is, where (, , ) denotes standard local coordinates on  1 (R, R).The first-order jet bundle of maps R → R is a fibered manifold defined by the space of all 1-jets of smooth sections of the trivial bundle  : R × R → R. The 1-graph of a solution curve of differential equation ( 11) is a curve on S 0 , represented by a section   → (, (),   ()), on which contact form  2 =  −  vanishes.The one-form  2 is defined by the pullback of the canonical contact form  =  −  to S 0 , by natural inclusion.Accordingly, solutions of the exterior differential system with  1 = , are in one-to-one correspondence with the solutions of (11).For the vector fields on S 0 , we have the relation   (  ) =    and 1-graph of a solution curve of ( 11) is an integral curve of the vector field  1 .
Similar to ODEs, any first-order PDE with one dependent and two independent variables, in the form is described as a submanifold in the first-order jet bundle of maps R 2 → R. In the vicinity of a point, where /  ̸ = 0, Implicit Function Theorem states that one can write (15) as If   is known, say   = (, , ), then ( 16) may be written as an overdetermined system with (, , ) = (, , , (, , )), such that   =   must hold on an integral surface.The first-order jet bundle of maps R 2 → R is a fibered manifold defined by the space of all 1-jets of smooth sections of the trivial bundle  : R 2 × R → R 2 of two independent and one dependent variables and denoted by  1 (R 2 , R).Let (, , , , ) be the standard coordinates on  1 (R 2 , R).The 1-graph of a solution or an integral surface of ( 15) is represented by the section   =  and   =  on which the canonical contact form vanishes.That is, solutions of the exterior differential system are in one-to-one correspondence with the solutions of (17).Consider three-dimensional submanifold Σ →  1 (R 2 , R) determined by the equations  =  (, , ) ,  =  (, , ) . ( Let us denote the pullback of  to Σ by  3 = −(, , )− (, , ).In terms of the differential 1-forms defined on Σ with local coordinates (, , ), and the inclusion map, an integral submanifold S of ( 15) is determined by  : Since the exterior derivative commutes with the pullback,  *  3 = 0 holds also on this two-dimensional integral manifold.Thereof, the identity I =   −   +    −    must vanish identically on integral manifold S. Due to the Frobenius Theorem, the integrability condition is described by It is suitable to note here that I = 0 does not necessarily hold identically on Σ.
The vector fields satisfy the relation   (  ) =    .Integral curves of the vector fields  1 and  2 , passing through the point ( 0 ,  0 ) ∈ R 2 , are determined by solutions of the ordinary differential equations which correspond to characteristic equations for (17).Moreover, since  =   +   on an integral surface, parameter curves, passing through a point ( 0 ,  0 ) for ( 0 ,  0 ) ∈ R 2 , on a solution are determined by the solutions of (25).In the subsequent section, we will define Riemannian metrics on S 0 and Σ in terms of the exterior differential systems determined by the set of 1-forms { 1 ,  2 } and { 1 ,  2 ,  3 }.They define local coframe fields on S 0 and Σ, respectively, since both of them are linearly independent at each point on respective coordinate neighbourhoods.For detailed description of differential equations in jet bundle formalism we refer to [29,[31][32][33].

Metric Connections Associated with First-Order Differential Equations
3.1.Connection on S 0 .We shall define a Riemannian metric on a coordinate neighbourhood of S 0 by the sum of squares of differential 1-forms (13) as follows: The coframe field { 1 ,  2 } on S 0 is dual to the orthonormal frame of vector fields (14).A solution curve of ( 11) is attached to an integral curve of  1 = / + (/) and an integral curve of / to  = const.By definition (  ,   ) =   .It is suitable to note that Riemannian metric ( 26) is not uniquely defined and depends on the choice of the coframe.The coframe that we use here may be thought as "best representative" associated with a differential equation since solutions are determined by vanishing of the canonical contact form in jet bundle.Moreover, our aim is to relate geodesic curves of the metric connection to solution curves of given ordinary differential equation simply as ∇  1  1 = 0.In other words, an integral curve of ( 11) can be obtained as the projection of a geodesic curve.
In the formulation that we use, the one-forms   ,  = 1, 2 are defined by the restriction or pullback of the canonical contact form  =  −  and  on  1 (R, R) to S 0 .Since  = (,) on S 0 , in terms of a local coordinates (, ), coordinate description of the Riemannian metric is obtained as by substituting  1 =  and  2 = − in (26).Notice that the components of the Riemannian metric purely depend on the right hand side of a given ODE.
The structure equations for the coframe field ( 1 ,  2 ) are determined as From here, we can define unique o(2, R)-valued torsion-free connection as In terms of Θ, structure equations take the form where Accordingly, from definition (10) of the covariant differentiation, we obtain Then, it is concluded that 1-graph of a solution curve of (11) defines a geodesic curve of connection (29).
Curvature of the connection Θ is found as It is clear that K is exactly the Gaussian curvature of the metric compatible torsion-free connection, whose components are determined in local coordinates by the well-known formula: From ( 32) or (33), we see that the Gaussian curvature of the metric connection is given by As a consequence, we get the following.
Theorem 1. Gaussian curvature of the metric connection (29) on the surface corresponding to the equation / = (, ) is determined by the following inhomogeneous Burgers' equation: This representation for the curvature is worthwhile to word in some aspects: By means of expression (35), the inviscid and viscous Burgers' equations and also KdV equation can be seen as the equations for surfaces with certain Gaussian curvatures.Also, by virtue of our representation, the problem of the construction of Riemannian metrics associated with certain first-order ODEs turns out to finding a solution of certain Burgers' equations.Moreover, expression (35) for curvature is rather simple and hence it is very convenient for computational purposes in a way that it is calculated by the Lie derivative of the right hand side of a given ODE in the direction of the vector field  1 =   +  .Besides, the spaces of constant curvature, which admit isometry group of maximal dimension, can be attached to some ODEs by solving certain inhomogeneous Burgers' equations.
From (35) it follows that S 0 is flat iff (, ) satisfies for some smooth function ().This identity tells us that the characteristic curves of (36) correspond to the solution curves of differential equation (11).Therefore, the inviscid Burgers' equation corresponds to a flat surface in  1 (R, R) associated with the equation Also, it is geometrically reasonable to investigate a question like whether the right hand side of (36) depends also on , since it is ultimately related to obtaining a space of constant curvature K = 0, ±1 from a single ODE.These are in fact determined by the following Burgers' equations: +  () , such that the characteristic curves of these first-order PDEs correspond to solutions of the system of equations for (, ) = (), (, ) =  + (), and (, ) = − + (), respectively.Although solutions of these equations can be obtained as implicit functions of the corresponding first integrals by the method of characteristics, it is not hard to see the existence of such Riemannian metrics simply by the linear differential equation as follows: Proposition 2. A linear first-order equation describes a surface of constant curvature K = 0, ∓1 if and only if  satisfies the following differential equations: respectively.Moreover, in the case of K = 0, if () is an integrating factor of (41), then (, ) satisfies the inviscid Burgers' equation   +   = 0.
On the other hand, to see what happens with our representation if the viscosity is added, identity (35) for the Gaussian curvature tells us that viscous Burgers' equation represents a surface with curvature To see this, let us consider the ODE with (, ) = −2](ln )  .Then, as it is well known, (, ) satisfies the viscous Burgers' equation provided that (, ) satisfies the heat equation   = ]  .In this case, Gaussian curvature of the surface corresponding to (44) is determined by the differential equation which is equivalent to (45) due to (35).The existence of such curvature is guaranteed by the existence of a solution of the heat equation and so is (45).
Another interesting integrable model is Korteweg-de Vries (KdV) equation which admits soliton solutions and appears widely in the dynamics of shallow water waves [34].From (35), it is seen that (47) describes a surface with Gaussian curvature 3.2.Connection on Σ. Analogues to the previous subsection, we shall define a Riemannian metric on a coordinate neighbourhood of Σ by the sum of squares of the elements of the coframe { 1 ,  2 ,  3 } as Coframe { 1 ,  2 ,  3 } is dual to the orthonormal frame field of vector fields  In order to construct o(3, R)-valued torsion-free connection, let us consider the structure equations for the coframe { 1 ,  2 ,  3 }: where I =   −   +    −   .Construction of o(3, R)valued connection needs a little bit of work.To this end, we shall solve the system of equations Advances in Mathematical Physics By using Cartan's Lemma for the 1-forms , , and , we obtain Accordingly, we have where the connection matrix of 1-forms is defined by Therefore, we have the following: where , , and  are defined, respectively, by  = (1/2)I 3 ,  = (1/2)I 2 +    3 , and  = −(1/2)I 1 +    3 .
By definition (10) of the covariant differentiation, the covariant derivatives for the frame field ( 1 ,  2 ,  3 ) are deduced as where J = −(1/2)I.It follows from here that the vector fields  1 and  2 are covariantly constant along themselves with respect to the metric compatible connection.
As it is seen directly from (58) and (59), a PDE with one dependent and two independent variables defines codimension one distribution in three spaces.Let us denote this distribution by H. Since H is spanned pointwisely by the orthonormal vector fields ( 1 ,  2 ), the curvature of H is determined by the equation The following proposition states the intrinsic and extrinsic properties of an integral surface of (17).For the geometry of submanifolds in a Riemannian manifold, see, for instance, [35].
Proposition 4. Any integral surface of ( 17) is an intrinsically flat and totally geodesic submanifold of Σ.
Proof.Due to [ 1 ,  2 ] = −I 3 ,  3 defines a normal vector field of the integral surfaces of (17).Since  =    +    on an integral surface, Riemannian metric (51) reduces to Euclidean metric on an integral surface  = (, ).The intrinsic curvature of the surface  = (,) is determined by the single independent component of Riemannian curvature tensor associated with (62).Therefore, any integral surface defines intrinsically flat surface.The second fundamental form of an integral surface identically vanishes, since on an integral surface we have ∇  1  3 = ∇  2  3 = 0.It follows from here that any integral manifold of ( 17) is a totally geodesic submanifold of Riemannian manifold Σ.
Now, let us state and prove the theorem manifesting that the scalar curvature of the Riemannian manifold (Σ, ) is determined by the integrability condition for (17) and the Gaussian curvatures of the surfaces corresponding to the characteristic differential equations (25).
Theorem 5.The scalar curvature of the Riemannian manifold (Σ, ) constructed from the system of equations ( 17) is determined by where J = −(1/2)I and K 1 , K 2 are the Gaussian curvatures of the surfaces corresponding to equations in (25), respectively.
Proof.The proof is based on the straightforward calculation.