Properties of a closed solid klein bottle?

Question

Okay, so I'll denote the topological space of the klein bottle as $Kb$ (because I don't know what the proper notation is). I'm curious about a closed version of the solid klein bottle, which I believe would be represented by the cartesian product $Kb \times S^1$. 1: Can this manifold be embedded in 5 dimensional euclidean space? If not, what is the lowest dimensional space that it can be embedded in? 2: Has anyone studied a manifold like this and could you give me some sources? More specifically, has a manifold like this ever been parameterized with a position vector?

Answer 1

1) People often denote the Klein bottle by $K$. $K \times S^1$ is not what you mean by the closed version of the Klein bottle, since the 'closed version of the Klein bottle' should have boundary the Klein bottle, and your construction has no boundary. What you really want is to ape the construction of the Klein bottle as follows. To build the Klein bottle, you take $S^1 \times I$ and glue $S^1 \times \{0\}$ to $S^1 \times \{1\}$ by $(z,0) \sim (\bar z, 1)$. (That is, you glue them by a reflection.) Now do the exact same thing but replace $S^1$ by the unit disc $D^2$. This is what's known as the solid Klein bottle.

2) Yes, this can be embedded in $\Bbb R^5$. Actually, you can already embed it in $\Bbb R^4$.

3) People have studied most manifolds, including this. Can you be more precise about what you're looking for? (I'm not sure what you mean when you're talking about a position vector.)