Singularities of gradient vectorfields in [formula omitted

Abstract

Let N be a neighborhood of (0, 0,O) in 9 3 and V = U + W a C’ function, I > k + 3, from N to 9, where U is a homogeneous polynomial of degree k + 1, and W vanishes together with all its derivatives through order k + 1 at 0. Let g be a smooth Riemannian metric on .R3 and let X = grad, V. If k = 1, then the critical point of V at 0 is non-degenerate if and only if U,, U,,, and U, vanish together only at (0, 0, 0), in which case the singularity of X at (0, 0,O) is hyperbolic. Then the Hartman-Grobman theorem implies that X is topologically equivalent near (0, 0,O) to its linear part; i.e., the topological equivalence class of X is determined by its first non-zc:o jet. We denote this X Xi. For k > 1, let X, denote the homogeneous polynomial vectorfield of degree k with the same k-jet at (0, 0,O) as X (Xk is the “homogeneous part” of X). Let x denote the vectorfield on a neighborhood of S2 x (0) in S* X .5? obtaining from X by polar blowing up. The main result is: