A structure of the solenoidal 2D vector and 2-tensor fields given in a domain with the conformal Riemannian metric

The Helmholtz decomposition of a vector field on potential and solenoidal parts is much more natural from physical and geometric points of view then representations through the components of the vector in the Cartesian coordinate system of Euclidean space. The structure, representation through potentials and detailed decomposition for 2D symmetric m-tensor fields in a case of the Euclidean metric is known. For the Riemannian metrics similar results are known for vector fields. We investigate the properties of the solenoidal vector and 2-tensor two-dimensional fields given in the Riemannian domain with the conformal metric and establish the connections between the fields and metrics.


Preliminary definitions and constructions
The Helmholtz decomposition of a vector field on potential and solenoidal parts is well known [1], [2]. This decomposition is much more natural from physical and geometric points of view then representation through the components over coordinate systems in Euclidean space. A profound generalization of Helmholtz decomposition to a case of the symmetric tensor fields, given in a compact Riemannian manifold, was suggested in [3]. The structure, representation through potentials and detailed decomposition for 2D symmetric m-tensor fields in the case of the Euclidean metric was established in [4], [5]. In a case of the Riemannian metric similar results are known partially only for vector fields. We investigate here the properties of the solenoidal vector and symmetric 2-tensor fields given in the Riemannian domain with the conformal metric and establish the connections between the fields and metrics. The solenoidal fields of general type and special type depending on the scalar potentials are considered.
Let in Euclidean space R n the Cartesian rectangular coordinate system be given. Points of R n have the coordinates (x 1 , . . . , x n ). We use the symbol B = {x ∈ R n | (x 1 ) 2 + . . . + (x n ) 2 < 1} for the unit ball, and ∂B for the unit sphere.
We suppose the ball B to be the Riemannian domain with the metric The designation T m (B), (S m (B)) is used for a set of covariant (symmetric) tensor fields of rank m with components u i 1 ...im (x), v i 1 ...im (x), . . . It is easy to change the covariant components by contravariant (and back) with usage of the operators of raising and lowering indices, is fixed; the origin of a vector ξ is at the point x}, and the manifold ΩB = {(x, ξ) ∈ T B | |ξ| 2 ≡ g ij ξ i ξ j = 1} of tangent unit vectors (in metric g). The boundary ∂(ΩB) consists from two parts, where ν(x) is unit vector of outer normal to ∂B at a point x. Remind also that the components of a tensor field depend on a point x ∈ B, which is connected with the tangent space T x B.
For u ∈ T m the operator of symmetrization σ : T m → S m is defined by an equality where Π m is the group of all permutations of the set {1, . . . , m}. At the same time with L 2 (B)-spaces we use the spaces C ∞ (S m (B)), C ∞ 0 (S m (B)), and Sobolev spaces H k (S m (B)), H k 0 (S m (B)) and H k (∂ + ΩB), k is integer, k ≥ 0. The index 0 from below specifies the corresponding subspaces of functions or tensor fields vanishing together with their derivatives on the boundary ∂B. Below we omit the sign B of unit ball in the designations of sets and spaces of functions and tensor fields.
The operator of inner differentiation d : where ∇ is the Riemannian connection, and the components of the tensor field (∇u) expressed through the components of u as follows, The divergence δ : The formula (4) contains Christoffel symbols, where g ij are the contravariant components of fundamental tensor. The inner product is defined as follows, We remind that a symmetric m- An essential generalization of Helmholtz decomposition for a vector field into a case of the symmetric tensor fields, given in a compact Riemannian manifold, was suggested in [3]. The symmetric tensor field f ∈ H 1 (S m ) can be represented as a sum of the potential dv and the solenoidal f s fields, For the vector fields it is valid more detailed decomposition [1], [2]. Namely, every vector field w ∈ H 1 (S 1 ) is decomposed into the sum of the potential dφ, solenoidal v and harmonic dh vector fields, where h is harmonic in B function, ν is the vector of outer to the boundary ∂B normal, v ∈ H 1 (S 1 ), φ ∈ H 2 . The decomposition (6) is unique. Later in [4] it was established that in 2D case the solenoidal field v can be represented through potential ψ in the form similar to the representation of a potential field, v = d ⊥ ψ (an exact definition of d ⊥ will be given below). The condition ⟨v, ν⟩ | ∂B = 0 can be change then to the condition ψ | ∂B = 0.
We formulate the definitions of ray transforms here for 2D case only. The longitudinal ray transform (LRT) [3] over a symmetric m-tensor field f ∈ H 1 (S m ) is linear operator where is the value of the parameter (arc length) at which the geodesic intersects the circle ∂B for the second time. The boundary ∂ + ΩB is defined by (2). Alongside with LRT it is possible to define the transverse ray transform (TRT) over a symmetric m-tensor field f ∈ H 1 (S m ). Namely, TRT is the linear operator P ⊥ : H 1 (S m ) → H 1 (∂ + ΩB), The vectors η x,ξ (t) andγ x,ξ (t) satisfy to the conditions for one and the same value of the parameter t.
Finding the second component η 2 of η, we obtain for the contravariant components of the vector η, and for covariant. It is easy to see that Besides, we have connections for contravariant components of ξ, and for covariant components. The posed task can be solved also and with a help of discriminant tensor e with contravariant components e 12 = −e 21 = 1/ √ g, e 11 = e 22 = 0, and covariant e 12 = −e 21 = − √ g, e 11 = e 22 = 0.
In other words, Obviously the components of sought-for vector η, orthogonal to the given ξ, can be found as This expressions coincide with (9) obtained above. Accept the designation ξ ⊥ = η for the vector, orthogonal (in a sense of Riemannian metric) to the given ξ, |η| = |ξ|. Note two spacial cases of Riemannian metrics.

The solenoidal 2D vector and 2-tensor fields
We remind the definitions and properties of certain differential operators. Let φ, ψ be the functions of C k smoothness, k ≥ 0. An action of the operator of inner differentiation d : C k → C k−1 (S 1 ) on the function φ leads to (covariant) potential vector field u ∈ C k−1 (S 1 ), The operator d coincides in this case with the operator of gradient. Let us define the operator where e is discriminant tensor (10). The action of d ⊥ on ψ leads to (contravariant) vector field v ∈ C k−1 (S 1 ), For the covariant components of the field v = (d ⊥ ψ) we obtain The divergence δ : C k (S 1 ) → C k−1 , acting on the vector field v, yields a scalar, Here we use the properties g ij ;k = 0 of the metric tensor, and Γ i ji = ∂(ln √ g) ∂x j = 1 √ g ∂ √ g ∂x j of the Christoffel symbols. Direct calculations show that the field v is solenoidal, δv = 0. The operator δ ⊥ : C k (S 1 ) → C k−1 is defined as follows, δ ⊥ u = −δu ⊥ , where u ⊥ is the vector field, orthogonal to the field u (see (8), (9)). With usage of (14), (8) we obtain Certain properties and connections of vector, 2-tensor fields and the defined above differential operators are formulated below. Proposition 1.