Understanding an abstract notion of "frame" for Klein geometries

Question

The Wikipedia page for "Moving frame" introduces the following abstract notion of a "frame" for a Klein geometry:

Formally, a frame on a homogeneous space $G/H$ consists of a point in the tautological bundle $G → G/H$.

Based on the previous section(s) of the article, it seems as if this definition generalizes the notion of "linear frame", "orthonormal frame", "affine frame", "Euclidean frame", and "projective frame", but I'm having trouble understanding how to get "back down" to the concrete concepts from the abstract definition.

In particular, I would like to understand how I can use this abstract and coordinate-free definition of "frame" to express a certain "family" of intuitive and commonly-used ideas from my engineering domain in the appropriate mathematical formalism. This "family" of concepts has to do with "expressing" points (and transformations) with respect to frames, and doing type-checking-style reasoning on the resulting "expressed" objects. For example:

• You can "express" an abstract point in the homogeneous space $G/H$ in a chosen frame.
• You can only "compare" two points if they are "expressed" in the same frame.
• The group action of $G$ on the homogeneous space $G/H$ can be viewed (not uniquely) as a "change of frames".
• You can only apply the action of $g \in G$ to a point in $G/H$ if that point is "expressed in the correct frame".
• The elements of the Lie algebra $\mathfrak g$ for $G$ can be expressed in a given frame (for e.g. $G = SE(3)$ and $H = SO(3)$ this corresponds to choosing an orthonormal frame and expressing $X \in \mathfrak{se(3)}$ as an angular and linear velocity $(\omega^\top, v^\top)^\top$ rooted at the frame origin).
• For a given $x \in G$, the adjoint representation $\operatorname{Ad}_x : \mathfrak g \to \mathfrak g$ transforms Lie algebra elements from one frame to another (agreeing with the change-of-frames induced on $G/H$ itself by $x$).

Does the above (or any other) abstract notion of frame allow one to express this family of concepts formally? I have a preference for notions that don't rely on a notion of "choosing a basis" or anything else that depends on the "type" of the points in the homogeneous space. Alternatively, you may convince me that I'm asking the wrong question or that my assumptions about which notions can (or should) be mathematically formalized are incorrect :)

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For context, I am a robotics engineer with an M.S.-level academic math background who works with various Lie groups in applications and I'm generally interested in understanding the deeper mathematics of these objects. I already understand basic Lie theory and the definition of Klein space and how metric, affine, projective, spherical (etc.) geometries are all instantiations of this general notion. I don't have any prior experience trying to understand Klein geometry beyond the basic definitions.

Answer 1

My approach to this is as follows. Suppose we have a complicated (real-world) object $X$ which we can describe by means of a well known (mathematical-world) object $S$. Examples:

• A plane piece of land $X$ described by $\mathbb{R}^2$.
• A list of students $X=\{Antonio, Carlos,\ldots,Rodrigo\}$ described by the set $\{1,2,,\ldots,32\}$.
• ...

The object $X$ is homogeneous with respect to the group $Bij(S)$ in the sense that:

1. There is a physical linkage, or external linkage, from $S$ to $X$, that is introduced externally. Let's call it $\phi: S\to X$.
2. All the possible descriptions of $X$ are given by $P=\{\phi\circ g: g\in Bij(S)\}$.
3. Suppose we have a distinguished point in $s\in S$ (for example $(0,0)\in \mathbb{R}^2$ or $1\in \{1,\ldots,32\}$). At first it may seem that the point $x=\phi(s)\in X$ is a privileged point, since it is described by the "main point" of $S$. But it turns out that $$ P=\sqcup_{y\in X} P_y $$ being $P_y=\{\phi\circ h:h\in H\}$ (with $H$ the isotropy group of $s$ in $Bij(S)$) all the descriptions of $X$ "centered at" $y$. That is, $P_y$ consist of descriptions of $X$ in which $y$ is described by the distinguished point $s\in S$. It is clear that all $P_y$ are "equal" (conjugates), and so we say that $X$ is homogeneous. For example, no point of the land is special, all of them can be chosen to be the center of the world, a no student is special, all of them can be chosen to be associated with the label 1.

Now, it is interesting to now if we can shrink $Bij(S)$ to a subgroup $G$ in such a way that $X$ still conserve this property. Why do we want to restrict? Because there are descriptions which are more important than others. For example, any "crazy" bijection $g\in Bij(\mathbb{R}^2)$ my not give rise to an interesting description $\phi \circ g:\mathbb{R}^2\to X$. We are interested in bijections that do not clutter the set too much. The choice of $G$ is again an external data, which depends on the nature of the problem we are interested in. But the group needs to be transitive, in order to conserve homogeneity.

An element $\phi\circ g\in P$ is a $G$-basis or $G$-description of $X$ in the sense that we can define the coordinates of a point $y\in X$ as $$ (\phi \circ g)^{-1}(y)\in S $$

The group $G$ not only serves to parameterize the "$G$-descriptions" of $X$. It also defines an action on $X$ through $\phi:S\to X$. The action of $g\in G$ on $y\in X$ is $gy=\phi(g(\phi^{-1}(y)))$. If we fix a point $x\in X$ (corresponding to certain $s\in S$ by means of $\phi$) we get a bijection $\Psi:X\to G/H$ (being $H$ the isotropy group of $x$) given by $$ \Psi:y\mapsto g_y H $$ $$ gx \style{display:inline-block; transform:scale(-1,1);}{\mapsto} gH $$

being $g_y$ an element of $G$ carrying $x$ to $y$.

This bijection $\Psi$ let us to express the points of $G/H$ in "coordinates" in $S$. Given $\bar{g} \in G$, the element $\bar{g}H$ have coordinates $$ (\phi \circ g)^{-1}(\Psi^{-1}(\bar{g}H))=(\phi \circ g)^{-1}(\bar{g}x)= $$ $$ =(\phi \circ g)^{-1}(\phi(\bar{g}(\phi^{-1}(x))))=(g^{-1}\bar{g})\phi^{-1}(x)= $$ $$ =(g^{-1}\bar{g})s $$

In particular, if we think in $X=G/H$ described by the set $S=G/H$ by means of the identity and with selected points $s=x=H$, then an abstract point $\bar{g}H \in G/H$ is expressed in a frame $g\in G$ as $$ g^{-1}\bar{g}H $$ Example: $G=\mathbb{R}^2 \rtimes GL(2)$, $H=GL(2)$, $G/H=\mathbb{R}^2$, $x=s=\tau_{(0,0)}H$ An element $g=\tau \circ A\in G$ is a translation $\tau$ following a linear transformation $A$, and sends the canonical reference system $((0,0),(1,0),(0,1))$ to a new one. The coordinates of a point $y\in \mathbb{R}^2$ in the new reference system are computed by means of $A^{-1}\tau^{-1}y$.

Maybe this link can help you to see the point.