Is there any way to have a geometric, visual intuition of what this manifold could be?

Asked Dec 21 '23 at 11:23

Active Dec 21 '23 at 12:55

Viewed 57 times

Let's consider $H$ the space of $3\times3$ matrices with real coefficients of the form $$A =\begin{bmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ z & y & 1 \end{bmatrix}$$ with the following equivalence relation $A\sim A' \iff \exists\gamma\in H \text{ with } x,y,z \in \mathbb{Z} \text{ such that } A' = \gamma A$

From my understanding $H$ is essentially $\mathbb{R}^3$, and the relation basically translates to $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} \sim \begin{bmatrix} x+a \\ y+b \\ z+bx+c \end{bmatrix} \text{ with } a,b,c\in \mathbb{Z}$$

Now, is there any way to see $M= H/ \sim$ visually? (And, bonus question, to actually demonstrate it's a compact, topological manifold?)

edited Dec 21 '23 at 12:55

asked Dec 21 '23 at 11:23

ccnptr

Why do you write "the subspace $M$"? $M$ is a quotient, so does not immediately come with an embedding into any space, and definitely does not embed into $H$. – Sergey Guminov Dec 21 '23 at 11:26
You are right it's an error, I'm gonna correct it – ccnptr Dec 21 '23 at 11:30
3

Is there supposed to be a 1 in the bottom-right entry of $A$? If so, you can find a lot of useful keywords about this space in https://math.stackexchange.com/questions/434384/the-heisenberg-manifold – Eero Hakavuori Dec 21 '23 at 11:46
Yes! Sorry I wrote it kinda in a hurry and made quite a few mistakes! – ccnptr Dec 21 '23 at 12:54
For visualization, perhaps consult Jeff Weeks's The Shape of Space. – Andrew D. Hwang Dec 21 '23 at 14:36

Is there any way to have a geometric, visual intuition of what this manifold could be?

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