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.Mon, 08 Apr 2019 18:21:31 +0200Is SageManifolds adequate to work with homogeneous Riemannian manifolds?https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/I have just discovered the
[**SageManifolds Project**](https://sagemanifolds.obspm.fr/index.html),
which computes several objects from differential geometry. I admit I haven't
studied in details the tutorials, because before spending several hours
doing that I want to be sure that it is going to help me in my purpose.
I want to compute the Riemann curvature tensor of compact homogeneous Riemannian manifolds.
Roughly speaking, each of those spaces has the following ingredients:
1. The differential manifold is given by $M=G/K$, where $G$ is a compact
Lie group and $K$ is a closed subgroup of $G$.
2. At the Lie algebra level, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$
with $\mathfrak{p}$ an $\textrm{Ad}(K)$-invariant subspace of $\mathfrak{g}$. The
tangent space $T_{eK}G/K$ is naturally identified with $\mathfrak{p}$.
3. An $\textrm{Ad}(K)$-invariant inner product $\langle \cdot,\cdot\rangle$
on $\mathfrak{p}$.
Thus, the Riemannian metric on $G/K$ is obtained by translating the inner
product $\langle \cdot,\cdot\rangle$ on $T_{eK}G/K \simeq \mathfrak{p}$ to
any $T_{gK}G/K$ by the map $xK\mapsto gxK$ (which becomes an isometry).
The main point with these spaces is that the whole geometry is determined by
$\mathfrak g$ and $\langle\cdot,\cdot\rangle$, and it is not necessary to
deal with charts.
In particular, one determines any curvature object (Riemann curvature tensor,
Ricci tensor, Scalar curvature, etc) only at the point $eK$.
In most of the examples that I quickly see in the tutorials of SageManifolds
begins by defining charts.
How can I work on a homogeneous Riemannian manifold without defining charts?
It would be very useful to count with a simple example, say $G=SU(2)$,
$K=\{1\}$, and the inner product on $\mathfrak p=\mathfrak g$ has orthonormal
basis $\{aX_1,bX_2,cX_3\}$ where $a,b,c$ are positive numbers and
$$
X_1 = \begin{bmatrix} i & 0 \newline 0 & -i \end{bmatrix},
\qquad
X_2 = \begin{bmatrix} 0 & 1 \newline -1 & 0 \end{bmatrix},
\qquad
X_3 = \begin{bmatrix} 0 & i \newline i & 0 \end{bmatrix}.
$$Mon, 08 Apr 2019 11:47:03 +0200https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/Answer by eric_g for <p>I have just discovered the
<a href="https://sagemanifolds.obspm.fr/index.html"><strong>SageManifolds Project</strong></a>,
which computes several objects from differential geometry. I admit I haven't
studied in details the tutorials, because before spending several hours
doing that I want to be sure that it is going to help me in my purpose.</p>
<p>I want to compute the Riemann curvature tensor of compact homogeneous Riemannian manifolds.
Roughly speaking, each of those spaces has the following ingredients:</p>
<ol>
<li><p>The differential manifold is given by $M=G/K$, where $G$ is a compact
Lie group and $K$ is a closed subgroup of $G$.</p></li>
<li><p>At the Lie algebra level, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$
with $\mathfrak{p}$ an $\textrm{Ad}(K)$-invariant subspace of $\mathfrak{g}$. The
tangent space $T_{eK}G/K$ is naturally identified with $\mathfrak{p}$.</p></li>
<li><p>An $\textrm{Ad}(K)$-invariant inner product $\langle \cdot,\cdot\rangle$
on $\mathfrak{p}$.</p></li>
</ol>
<p>Thus, the Riemannian metric on $G/K$ is obtained by translating the inner
product $\langle \cdot,\cdot\rangle$ on $T_{eK}G/K \simeq \mathfrak{p}$ to
any $T_{gK}G/K$ by the map $xK\mapsto gxK$ (which becomes an isometry).</p>
<p>The main point with these spaces is that the whole geometry is determined by
$\mathfrak g$ and $\langle\cdot,\cdot\rangle$, and it is not necessary to
deal with charts.</p>
<p>In particular, one determines any curvature object (Riemann curvature tensor,
Ricci tensor, Scalar curvature, etc) only at the point $eK$.</p>
<p>In most of the examples that I quickly see in the tutorials of SageManifolds
begins by defining charts.</p>
<p>How can I work on a homogeneous Riemannian manifold without defining charts?</p>
<p>It would be very useful to count with a simple example, say $G=SU(2)$,
$K={1}$, and the inner product on $\mathfrak p=\mathfrak g$ has orthonormal
basis ${aX_1,bX_2,cX_3}$ where $a,b,c$ are positive numbers and
$$
X_1 = \begin{bmatrix} i & 0 \newline 0 & -i \end{bmatrix},
\qquad
X_2 = \begin{bmatrix} 0 & 1 \newline -1 & 0 \end{bmatrix},
\qquad
X_3 = \begin{bmatrix} 0 & i \newline i & 0 \end{bmatrix}.
$$</p>
https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/?answer=46048#post-id-46048In the current setting, all calculations on manifolds end as calculations on charts. Nothing specific to homogeneous manifolds has been implemented yet. Would you be interested in developing such a part? Note that the implementation of Lie groups has started, see [here](http://doc.sagemath.org/html/en/reference/groups/sage/groups/lie_gps/nilpotent_lie_group.html).Mon, 08 Apr 2019 13:24:42 +0200https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/?answer=46048#post-id-46048Comment by emiliocba for <p>In the current setting, all calculations on manifolds end as calculations on charts. Nothing specific to homogeneous manifolds has been implemented yet. Would you be interested in developing such a part? Note that the implementation of Lie groups has started, see <a href="http://doc.sagemath.org/html/en/reference/groups/sage/groups/lie_gps/nilpotent_lie_group.html">here</a>.</p>
https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/?comment=46052#post-id-46052Well, I could try to help with the mathematical background, but my program skills are pretty poor, so somebody should optimize every algorithmMon, 08 Apr 2019 18:21:31 +0200https://ask.sagemath.org/question/46046/is-sagemanifolds-adequate-to-work-with-homogeneous-riemannian-manifolds/?comment=46052#post-id-46052