By S. T. Hu
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Thus V and R can be considered as instances of the Sent Venant operator. 4. 2 33 Each of these three operators has its own advantages. 1), as we shall see later. The operator R has the most of symmetries (which are similar to the symmetries of the curvature tensor ) and, consequently, is more appropriate for answering the question about a number of linearly independent equations in the system W f = 0. 17). jm−1 : (i1 . . im )(j1 . . 8) there exists a unique solution to the equation σ(i1 .
Im ) with respect to the index im . 1). jm = α(im jm ) σ(i1 . . im−1 ) σ(j1 . . im−1 . 2 For f ∈ C ∞ (S m ), the next relations are valid: 1 σ(i1 . . im ) σ(j1 . . jm . 5) 32 CHAPTER 2. THE RAY TRANSFORM ON EUCLIDEAN SPACE P r o o f. From the definition of V and the evident equality σ(i1 . . im ) σ(j1 . . jm ) α(im jm ) σ(i1 . . im−1 ) σ(j1 . . jm−1 ) = = σ(i1 . . im ) σ(j1 . . jm ) α(im jm ), we obtain m−1 1 m−1 σ(i1 . . im ) σ(j1 . . jm = σ(i1 . . im ) σ(j1 . . im . 4). Decomposing the symmetrizations σ(i1 .
9). The lemma is proved. 2 we need the next easy 34 CHAPTER 2. jq : i1 . . ip (j1 . . jq ). jk ) ∈ T p+k (k = 0, . . 12) has at most one solution z ∈ C q+1 (T p ; U ) satisfying the initial conditions ∇k z(x0 ) = z k (k = 0, 1, . . , q − 1). jq ; jq+1 : i1 . . ip (j1 . . jk : i1 . . ip (j1 . . jk ) (k = 0, 1, . . , q − 1). 13). jq : (i1 . . ip )(j1 . . jk : (i1 . . ip )(j1 . . jk ) (k = 0, 1, . . 13) belongs to C q+1 (S m ; U ). For p = 0, q = 1 the lemma is equivalent to the claim of necessity and (in the case of a simply-connected U ) sufficiency of the conditions ∂yi /∂xj = ∂yj /∂xi for integrability of the Pfaff form yi dxi .