I am asking this question to clarify a comment that appears below Eq.(32) of this paper, which applies Morse theory to classify van Hove singularities in energy bands of crystalline solids. These energies are real (and smooth) functions of momentum, and the space of momenta of a $d$-dimensional crystal has the topology of a $d$-dimensional torus.
To recap the basics:
The Poincaré-Hopf theorem relates the index of isolated zeros of a differentiable vector field $v$ on $M$ to the Euler characteristic $\chi(M)$, namely $$\sum_i\textrm{index}_{x_i}(v)=\chi(M),$$ where $x_i\in M$ are the locations of the isolated zeros. (EDIT: Here it is assumed that all zeros of $v$ are isolated.)
Specifically for $d$-dimensional torus $T^d$ (i.e. a product of $d$ copies of $S^1$) one can use the above relation to easily verify that $$\chi(T^d)=0.$$ This is because as $v$ on $T^d$ one can take a constant vector field that winds along one of the constituent circles of the torus, and this clearly has no zeros.
The general question:
I am wondering about the implications of the vanishing $\chi(T^d)=0$ for Morse theory on $T^d$; in particular, whether it enforces the existence of saddle points for smooth functions $f$ on $T^d$. To be more precise, by taking a (suitably differentiable) function $f$ on $M$, a vector field $v$ can be obtained simply as the gradient of $f$. (There are presumably some restrictions on $f$ such that the gradient only exhibits zeros at isolated points of $M$.) This turns critical points of $f$ (i.e. minima, maxima, saddles) to isolated zeros of a vector field.
For simplicity, let me limit attention to only the ordinary critical points. Then the function $f$ can be expanded near a critical point $x_0$ as $f(x) = f(x_0) + \sum_{ab} H_{ab}(x_0) x_a x_b$ where $H(x_0)$ is the Hessian matrix computed at $x_0$, and $x_a$ are local coordinates centered at the critical point. The index of such a critical point (which translates to the index of the isolated zero in the gradient of $f$) should, quite generally, be $$\textrm{index}_{x_0} (f)=\textrm{sign} \det H(x_0).$$
One should also note that any bounded function $f$ necessarily has at least one minimum and at least one maximum.
For $d=2$, one finds that both minima and maxima of $f$ have index $+1$, whereas saddles have index $-1$. Since any bounded function must have at least one minimum and one maximum, which together have index $+2$, the Poincaré-Hopf theorem implies the presence of additional critical points with index $-2$. Assuming there are only the ordinary critical points, the compensation is achieved by the presence of two saddle points. A very simple function such as $\cos x + \cos y$ (with $x,y \in [0,2\pi]$) provides an example with such critical points.
For $d=3$, however, the minima and maxima exhibit opposite index, i.e., their net index is already zero. Therefore, the Poincaré-Hopf theorem seems to imply that no saddle points (whose admissible index values in this dimension are either $+1$ or $-1$) are necessary. Indeed, the remark in the above referenced paper suggests that this is indeed the case. However, I have troubles constructing an explicit example of $f$ on $T^3$ that has no critical points beyond a single minimum and a single maximum.
Which brings me to my very concrete question:
Is it possible to find a concrete example of a smooth function $f$ on $T^3$ whose only critical points are a single minimum and a single maximum? If not, are there some additional constraints that must enter this kind of Poincaré-Hopf-Morse analysis?
You are misstating things a little bit: To begin with for P-H formula to apply, you have to have all zeroes to be isolated. As for your last question: No, every manifold admitting such a function is homeomorphic to the sphere, see this question, including comments.
What critical points of a Morse function really "count" are handles in a handle decomposition of a manifold, which provides a much more subtle information than $\chi$ alone.