chapter  4
18 Pages

General Method of Reconstruction

Let X, Σ be manifolds of dimension n > 1, Z be a smooth hypersurface in X × Σ, such that the natural projections pX : Z → X, and pΣ : Z → Σ have rank n. It follows that Z (σ) + p−1Σ (σ) is a smooth hypersurface in X for any σ ∈ Σ and Z (x) = p−1X (x) is a smooth hypersurface in Σ. A real smooth function Φ defined on a neighborhood of Z , such that Z = Φ−1 (0) and dΦ 6= 0 on Z will be called generating function. These conditions imply that dxΦ|Z = − dσΦ|Z 6= 0. Suppose that

(i) map

DX : Z × R+ → T ∗0 (X) + {(x, ξ) ∈ T ∗ (X) , ξ 6= 0}

is a diffeomorphism, where DX (x, σ, t) = (x, tdxΦ (x, σ)) . This condition implies that pX is a proper map and for any x ∈ X, Z (x) is diffeomorphic to a (n− 1)-sphere that meets each ray {ξ = tξ0, t > 0} in one point.