On the minimal dimension of the orbits of a $\mathbb R^n$-action

TL;DR

This paper investigates the minimal possible dimensions of orbits under smooth $R^n$-actions on manifolds, establishing bounds related to the manifold's dimension, rank, and Pontrjagin classes.

Contribution

It provides new bounds on orbit dimensions for $R^n$-actions, linking topological invariants like Pontrjagin classes to orbit size constraints.

Findings

Existence of orbits with dimension less than (m+k)/2.

Presence of non-zero Pontrjagin classes implies smaller orbits.

Bounds depend on manifold's dimension, rank, and Pontrjagin class degrees.

Abstract

Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a consequence, one shows that if there is a non-zero element of the ring of Pontrjagin classes of $M$ of degree $4\ell\geq 4$, then there exists an orbit of the action of dimension $\leq m-\ell-1$.