A traditional dealing with a semi-classical limit and Hopf theorem

This paper deals with a semi-classical limit(Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussing of it may be found in [7] 1 A semi-classical limit theorem Let M be a compact, closed Riemannian manifold of dim n, and V a vector field without degenerate zeros on M . Let Λ∗(M) be the space of differential forms on M , and D = d+ δ : Λ∗(M) → Λ∗(M) be the de Rham-Hodge operator, which is an elliptic operator. Let us consider a Witten’s deformation of d+ δ. Dt = (d+ δ) + t[V ∗ ∧+i(V )] : Λ∗(M) → Λ∗(M), where V ∗ is a 1-form dual to the vector field V , and V ∗∧ means the exterior product by V ∗, while i(V ) the interior product by V . Let 2t = D 2 t : Λ ∗(M) → Λ∗(M), and e−τ2t be the solution operator of the heat operator ∂ ∂τ + 2t. It is well known that e−τ2t is an integral operator, i.e. there exists a unique family of linear maps G(τ, q, p, t) : Λp(M) → Λq(M) such that (e−τ2tφ)(q) = ∫ M G(τ, q, p, t)φ(p)dp, ∀φ. Such a family of G(τ, q, p, t) is called a fundamental solution of the heat operator ∂ ∂τ + 2t. The fundamental solution can be determined by the following equations