Smooth Structure on the Moduli Space of Instantons of a Generic Vector Field

This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons of a generic smooth vector field. The result is stated in Theorem 1.4. CONTENTS 1. The results 1 2. Some basic ODE 7 3. Elementary differential topology of smooth manifolds with corners 9 4. Proof of the main theorem 10 References 19 1. THE RESULTS Let M be a smooth closed manifold and X : M → TM a smooth vector field, i.e. a section X : M → TM in the tangent bundle. The set X (X) of rest points of X consists of points of M where the vector field vanishes, X (X) := {x ∈M |X(x) = 0}. For any x ∈ X (X) the differential DX of the smooth mapX defines the endomorphism Dx(X) : Tx(M)→ Tx(M) called the linearization of X at x 1. The rest point x ∈ X (X) is called hyperbolic if the eigenvalues of Dx(X), {λ ∈ SpectDx(X)}, are complex numbers with real part <λ 6= 0. In particular Dx(X) is invertible. The hyperbolic rest point x ∈ X (X) is called of Morse type if one can find coordinates (u1, u2, · · · , un) in the neighborhood of x such that X = ∑ i±∂/∂ui. Given a hyperbolic rest point x ∈ X (X) the cardinality of the set of eigenvalues counted with multiplicity whose real part is positive is called Morse Index and is denoted by ind(x), ind(x) := ]{λ ∈ SpectDx(X)|<λ > 0}. A trajectory of X is a smooth path γ : R → M so that dγ dt = X(γ(t)) 2. One denotes by γy, y ∈M, the unique trajectory which satisfies γy(0) = y. 1991 Mathematics Subject Classification. 57R20, 58J52. partially supported by NSF grant no MCS 0915996. Dx(X) is defined as follows. Choose an open neighborhood U of x in M and a trivialization of the tangent bundle above U , θ : TU → U × Tx(M), with θ|Tx(M) = id. Consider Y := pr2 ·θ ·X : U → Tx(M) with pr2 the projection on the second component; Y (x) = 0. Observe that Dx(Y ) is independent of θ and define Dx(X) := Dx(Y ). 2If M is not compact R should be replace by an open interval, the maximal domain of the trajectory; when M is compact this domain is R. 1