Consider the $n$-dimensional Torus $T^n = \prod_{i=1}^n S^1$ as a subset of $\mathbb R^{2n} = \prod_{i=1}^n \mathbb R^2$ in the standard way.

Is it true that a generic $n$-multilinear functional on $\mathbb R^2$, i.e. a tensor in $T_0^n(\mathbb R^2)$, defines a Morse function on $T^n$? If so, does a generic element have the minimal number of critical points $2^n$ (among Morse functions)?

Generic here refers to all but a Haar null set of exceptions (dense would also be fine). The statement is reasonable in my opinion since these multilinear functionals seem to be good "height functions" in this context and the statement is indeed true for $n=1,2$. For $n=1$ it is obviuously true for any nontrivial linear function on $\mathbb R^2$. For $n=2$ we can write any $\omega \in T_0^n(\mathbb R^2)$ as $$ \omega = a\, e_1^* \otimes e_1^* + b\, e_1^* \otimes e_2^* + c\, e_2^* \otimes e_1^* + d\, e_2^* \otimes e_2^* $$ with respect to the standard basis of $\mathbb R^2$. Define the function $f : \mathbb R \times \mathbb R \to \mathbb R$ by $f(s,t) = \omega(e^{is},e^{it})$ and solve for critical points. For generic $a,b,c,d$ it follows rather quickly (modulo calculation errors) using the addition formulas for sine and cosine \begin{align*} s & = \frac{1}{2}\arctan \frac{c+b}{a-d} - \frac{1}{2}\arctan \frac{b-c}{a+d} + m\pi\ , \\ t & = \frac{1}{2}\arctan \frac{c+b}{a-d} + \frac{1}{2}\arctan \frac{b-c}{a+d} + n\pi \ , \end{align*} for $m,n \in \mathbb Z$. This has exactly four solutions $(s,t) \in (-\pi,\pi]^2$. Similarly one can check that for generic $a,b,c,d$, the Hessian matrix of $f$ at critical points $(s,t)$ is nondegenerate.

It is interesting to note that the statement can't hold for all $\omega \neq 0$ because if the coordinates of $\omega$ are for example given by $b = c = 0$ and $a = d = 1$, then the associated function $f(s,t) = \cos(s-t)$ has a continuum of critical points.

Preferably there is an argument that avoids calculations as above for $n > 2$ as this seems to be quite a laborious task, unless I miss something (which is likely). Maybe this question has been considered before or follows directly from more general principles available in the literature, but I wasn't able to find something. Any references in this direction are welcome.

  • $\begingroup$ Assuming your torus has the form $\mathbb R^n/\mathbb Z^n$, then multlinear maps with integer coefficients give well-defined maps from $T^n$ to $\mathbb R$. Are you referring to those? Then what does "generic" mean in this context? $\endgroup$ – Sebastian Goette Feb 26 '20 at 21:48
  • $\begingroup$ I am refering to multilinear maps $\omega : \mathbb R^2 \times \cdots \times \mathbb R^2 \to \mathbb R$ and its restriction to the "standard embedding" $T^n = S^1 \times \cdots \times S^1 \subset \mathbb R^2 \times \cdots \times \mathbb R^2$. Generic is with respect to the Haar measure on the vector space of such multilinear functions. $\endgroup$ – rozu Feb 27 '20 at 11:58
  • $\begingroup$ Thanks for the clarification - I should have read the question more carefully. But anyway - could one use the maps I referred to as well? $\endgroup$ – Sebastian Goette Feb 28 '20 at 9:45
  • $\begingroup$ I think the functions on $\mathbb R^n$ that lead to the same class of functions on $T^n = \mathbb R^n /(2\pi \mathbb Z)$ are finite sums of functions of the form $f(x_1,\dots,x_n) = r\cos(x_1 + a_1)\cdots\cos(x_n + a_n)$ for some $r, a_1,\dots, a_n \in \mathbb R$. Don't know if these functions have something to do with those you are suggesting. $\endgroup$ – rozu Feb 28 '20 at 16:02

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