# Revision history [back]

### Is SageManifolds adecuate to work with homogeneous Riemannian manifolds?

I have just discovered SageManifolds Project, which computes several objects from differential geometry. More info here. I admit I haven't studied in details the tutorials, because I want to be sure after losing several hours 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 $$X_1 = \begin{bmatrix} i & 0 \ 0 & -i \end{bmatrix}, \qquad X_2 = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}, \qquad X_3 = \begin{bmatrix} 0 & i \ i & 0 \end{bmatrix}.$$

 2 None slelievre 16479 ●19 ●147 ●328 http://carva.org/samue...

### Is SageManifolds adecuate to work with homogeneous Riemannian manifolds?

I have just discovered SageManifolds Project, which computes several objects from differential geometry. More info here. I admit I haven't studied in details the tutorials, because I want to be sure after losing several hours 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 $$X_1 = \begin{bmatrix} i & 0 \ 0 & -i \end{bmatrix}, \qquad X_2 = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}, \qquad X_3 = \begin{bmatrix} 0 & i \ i & 0 \end{bmatrix}.$$

 3 None slelievre 16479 ●19 ●147 ●328 http://carva.org/samue...

I have just discovered the SageManifolds Project, , which computes several objects from differential geometry. More info here. I admit I haven't haven't studied in details the tutorials, because before spending several hours doing that I want to be sure after losing several hours that it is going to help me in my purpose. purpose.

I want to compute the Riemann curvature tensor of compact homogeneous Riemannian manifolds. 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 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 $\mathfrak p$ $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ with $\mathfrak{p}$ an $\textrm{Ad}(K)$-invariant subspace of $\mathfrak g$. The $\mathfrak{g}$. The tangent space $T_{eK}G/K$ is naturally identified with $\mathfrak p$.$\mathfrak{p}$.

3. An $\textrm{Ad}(K)$-invariant inner product $\langle \cdot,\cdot\rangle$ on $\mathfrak p$. \cdot,\cdot\rangle$on$\mathfrak{p}$. Thus, the Riemannian metric on$G/K$is obtained by translating the inner inner product$\langle \cdot,\cdot\rangle$on$T_{eK}G/K \simeq \mathfrak p$to \mathfrak{p}$ 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 by $\mathfrak g$ and $\langle\cdot,\cdot\rangle$, and it is not necessary to deal with charts. charts.

In particular, one determines any curvature object (Riemann curvature tensor, 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 SageManifolds begins by defining charts. 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 $G=SU(2)$, $G=SU(2)$, $K={1}$, and the inner product on $\mathfrak p=\mathfrak g$ has orthonormal orthonormal basis ${aX_1,bX_2,cX_3}$ where $a,b,c$ are positive numbers and and $$X_1 = \begin{bmatrix} i & 0 \ \newline 0 & -i \end{bmatrix}, \qquad X_2 = \begin{bmatrix} 0 & 1 \ \newline -1 & 0 \end{bmatrix}, \qquad X_3 = \begin{bmatrix} 0 & i \ \newline i & 0 \end{bmatrix}.$$