Morse index of saddle equilibria of gradient-like flows on connected sums of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

TL;DR

This paper investigates the Morse index of saddle equilibria in gradient-like flows on certain connected sums of spheres, showing that under specific conditions, saddle indices are restricted to 1 or n-1, excluding intermediate values.

Contribution

It establishes a restriction on the Morse indices of saddle points in gradient-like flows on connected sums of spheres, assuming no heteroclinic intersections.

Findings

Morse index of saddles is either 1 or n-1.

No saddles with Morse indices between 2 and n-2 occur under the given conditions.

Flow dynamics are constrained by the topology of the underlying manifold.

Abstract

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.