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Keywords. Symbolic calculus, fundamental solution, curvature, local index. 1. Introduction In this paper we give, by means of symbolic calculus of pseudo-diﬀerential operators, both an extension theorem of a local version of the Gauss-Bonnet-Chern theorem given in  and that of a local version of the Riemann-Roch theorem given in . Let M be a Riemannian manifold of dimension n without boundary. The Gauss-Bonnet-Chern theorem is stated as follows: n (−1)p dim Hp (M ) = Cn (x, M )dv, M p=0 where Hp is the set of harmonic p-forms, Cn (x, M )dv is the Euler form if n is even and Cn (x, M )dv = 0 if n is odd.

Let λ0 ∈ Λα and D ∈ G∧ . For simplicity, we use the notation D = π∧,max D, V = VK∧,λ0 , K = π∧,max K∧,λ0 and πK,D = π ˆK∧,λ0 ,D . Suppose Ω− (D) ∩ V = ∅. Since Ω− (D) and V are closed sets, there are a neighborhood U of V and a constant R > 0 such that if > R then κ−1 D ∈ U. 2 the ray through λ0 is a ray of minimal growth for A∧,D . 4 ) is satisﬁed. Suppose Ω− (D) ∩ V = ∅ and let D0 ∈ Ω− (D) ∩ V . Since D0 ∈ V , we have D0 ∩ K = {0}. B. Gil, T. A. Mendoza as k → ∞. Note that for k large we have m k λ0 ∈ res A∧,D , so λ0 ∈ res A∧,κ−1 D k and therefore, Dk ∈ V .

6) Let Λ be a closed sector in C\R+ containing the half-plane { λ < 0}. 2 implies that every closed extension ∆∧,D , D ∈ G∧ , of the model Laplacian admits Λ as a sector of minimal growth. Equivalent geometric condition We identify G∧ with the Grassmannian Grd (E∧,max ) where d = − ind(A∧,min −λ) ◦ for λ ∈ Λα ⊂ bg-res A∧ . Let d = dim K∧,λ . 4 ). 28]. The following theorem states that the condition is also necessary. For D ∈ Grd (E∧,max ) let Ω− (D) = D ∈ Grd (E∧,max ) : ∃ { k ∞ k }k=1 ⊂ R+ such that → ∞ and κ−1 D → D as k → ∞ .