I've been messing with some computational topology texts and want to do some manifold/surface reconstruction on various data sets. I'd like to test on some purely geometric examples. Are there any explicit examples of compact $n$-manifold embeddings in $\mathbb{R}^m$, $m>n\ge3$, beyond spheres and tori? I've not found much with my search engine skills.
I'll admit I'm not terribly expert with theoretical geometry/topology; more of an applied mathematician, so please forgive my ignorance.