ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 10 Jul 2020 06:13:31 +0200Defining $S^1$ with four chartshttps://ask.sagemath.org/question/52400/defining-s1-with-four-charts/Hello,
I am trying to implement the construction of the manifold $S^1$ using the four charts, as described at en.wikipedia.org/wiki/Manifold#Circle . It's not clear to me how to define the charts in terms of the projection operators within SageMath. I'm also not sure how to isolate open sets that I would like to define the charts over. The circle is parameterized using polar coordinates, and the four charts have the form:
$
\begin{align*}
\varphi_1(r, \theta) &= \theta && \theta \in (-\frac{\pi}{3}, \frac{\pi}{3}) \\
\tag{1}
\end{align*}
$
$
\begin{align*}
\varphi_2(r, \theta) &= \theta && \theta \in (\frac{\pi}{6}, \frac{5\pi}{6}) \\
\tag{2}
\end{align*}
$
$
\begin{align*}
\varphi_3(r, \theta) &= \theta && \theta \in (\frac{2\pi}{3}, \frac{4\pi}{3}) \\
\tag{3}
\end{align*}
$
$
\begin{align*}
\varphi_4(r, \theta) &= \theta && \theta \in (\frac{7\pi}{6}, \frac{11\pi}{6}) \\
\tag{4}
\end{align*}
$
Once we have these charts, how would I define the four transition functions to complete the description?
Thank you for your help!wintermuteFri, 10 Jul 2020 06:13:31 +0200https://ask.sagemath.org/question/52400/Inverse of the transition map on a Manifold doesn't holdhttps://ask.sagemath.org/question/48264/inverse-of-the-transition-map-on-a-manifold-doesnt-hold/I was trying to represent S³ as a three dimensional manifold, with coordinates (x,y,z,w) in R⁴, and make the transition map from the upper cap w > 0 to the lateral cap z<0, with the charts being the graphs of the caps as functions. I came up with the following code:
M = Manifold(3, 'S^3')
N = M.open_subset('N')
projN.<x,y,z> = N.chart()
E = M.open_subset('E')
projE.<x,y,w> = E.chart()
ProjNE = projN.transition_map(projE,
[x,y, sqrt(1-x^2-y^2-z^2)], intersection_name='D',
restrictions1= z < 0, restrictions2= w>0)
It sounds reasonable, but calling
ProjNE.inverse()
failed. No problem, i tried using
ProjNE.set_inverse(x,y, -sqrt(1-x^2-y^2-w^2))
but i got the following warning:
Check of the inverse coordinate transformation:
x == x *passed*
y == y *passed*
z == -abs(z) **failed**
x == x *passed*
y == y *passed*
w == abs(w) **failed**
NB: a failed report can reflect a mere lack of simplification.
i don't know why the test is failing. The math sounds ok, where did it go wrong?JGCThu, 10 Oct 2019 22:07:30 +0200https://ask.sagemath.org/question/48264/