The 6-torus, obtained by identifying the boundaries of a parallelpiped of $\mathbb{R}^6$ (or, more properly, $\mathbb{C}^3$), is a Calabi-Yau manifold.  

You may find [this review paper](https://homes.psd.uchicago.edu/~sethi/Teaching/P484-W2004/calabi-yau.pdf) useful, though I wrote it as a graduate student almost 20 years ago and there are probably inaccuracies.  In addition, there may have been discoveries in the intervening years that are not reflected there.  Caveat lector.