I am curious what the object you obtain is when you cut the $d$-dimensional flat torus $\mathbb{T}^d :=\left(\mathbb{R} / \mathbb{Z}\right)^d$ along certain hyper-planes. Specifically, if you identify it with $[0,1)^d$ and consider a typical element $x=(x_1,\dots,x_d)$, I would like to cut it along the planes $x_i=x_j, i,j=1,\dots,d, i\neq j$. For $d=2$, one can check by hand that the resulting shape is isometric to $\mathbb{T}_{\sqrt{2}} \times [0,\frac{1}{\sqrt{2}}]$, where $\mathbb{T}_{\sqrt{2}}= \mathbb{R} /\sqrt{2}\mathbb{Z}$ is the appropriately scaled $1$-dimensional torus. Is there a general representation of what such a construction looks like in higher dimensions?
Edit: Just to clarify, I want to cut the torus along all such planes of which there are $d^2-d$ in dimension $d$. The final object (which will be a collection of manifolds of the same dimension $d$) may not be connected.
I don't know whether this helps but here are some pictures for the 4D case. The 4D torus I'm considering can be parameterized as $$ (a, b, c) \mapsto \left\{ \begin{align*} x &= (R_1 + (R_2 + R_3\cos\,a)\cos\,b)\cos\,c\\ y &= (R_1 + (R_2 + R_3\cos\,a)\cos\,b)\sin\,c\\ z &= (R_2 + R_3\cos\,a)\sin\,b\\ w &= R_3\sin\,a \end{align*}\right. $$ When slicing such a 4D torus in a hyperplane, we obtain shapes like this: