Traversally generic & versal vector flows: semi-algebraic models of tangency to the boundary

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal{V}^{\dagger} (X)$ and $\mathcal{V}^{\ddagger} (X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal{V}(X)$ of all fields (see Theorems 3.4 and Theorem 3.5). The definitions of boundary generic and traversally generic vector fields v are inspired by some classical notions from the singularity theory of smooth Bordman maps [Bo]. Like in that theory (cf. [Morin]), we establish local versal algebraic models for the way a sheaf of v-trajectories interacts with the boundary $\partial X$. For fields from the space $\mathcal{V}^{\ddagger} (X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2 \cdot \dim(X)$.

Keywords

vector fields, manifolds with boundary, singularity theory

2010 Mathematics Subject Classification

37C10, 57R25, 57R45

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Published 16 March 2017