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Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitelymany 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 offinitely many hyperbolic equilibria and their invariant manifolds intersectstransversally. We prove that if invariant manifolds of distinct saddles of agradient-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 $f^t$ itsMorse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$,so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

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