Update
Denote $\mathcal{M}$ by the manifold with boundary (i.e., solid torus). Let $i(x)$ denote the vector field index of the gradient flow $f$ of the Morse function, where $f$ points outward at all boundary points. If the point $q^*$ is the local minima of $-f$ then $i(q^*)=1$. According to the Poncar'e-Hopf theorem, $ 1+\sum_{x\in f^{-1}(0)-q^*}i(x) =\chi(\mathcal{M}) $
In our case (i.e., on solid torus), the index of $f$ on the critical points of Morse index 1 and 3 are $-1$, whilst the index of $f$ on the critical points of Morse index 0 and 2 are $+1$. Note that the Euler characteristic of solid torus is zero. According to $1+\sum_{x\in f^{-1}(0)-q^*}i(x) =\chi(\mathcal{M})$=0, one has that the vector field $f$ must has other critical points of Morse index 1 or 3 apart from local minima. Can we determine the specific Morse index of those critical points? For instance, if $f$ has extra one critical point apart from local minima, then can we determine whether its Morse index is 1 or 3?
Thanks at advance.