Question

I'd like to describe a "bottleneck" in n-dimensional topological vector space $\mathbb R^n$. For that I should probably try to define an "n-dimensional" cylinder.

I'd like to define something similar to a "hollow" cylinder but in say $\mathbb R ^n$ for n >= 2. I would probably define it as something homeomorphic to $S^{n-2} \times L$ where $L$ is a line crossing 0 (for a cylinder with finite volume I'd then assume this line to have two endpoints). Is there a name for this in Geometry or a reference where such a thing is discussed? Would this be the right way of defining a cylinder in n-dimensions? Any other suggestions?

Answer 1

I believe an abstract cylinder is any manifold having the form $M \times I$, where $I \subseteq \mathbb{R}$ is an interval. This is called a 'cylinder over $M$'. In your case, you'd want $M$ to have codimension 2 in the ambient space. Your definition is compatible with this.

Take a look at http://books.google.com/books?id=w7dActmezxQC&pg=PA158&lpg=PA158&dq=cylinder-over+manifold&source=bl&ots=4L8o9p3ECV&sig=0abI4PCpECfjaP0zXmltcxMaSeA&hl=en&ei=JFDYTvK-GeKWiQKr95SVCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDAQ6AEwAA#v=onepage&q=cylinder-over%20manifold&f=false for an example.