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Suppose we are given an oriented Riemannian manifold $S \subset \mathbb{R}^3$ (which I'll refer to as a surface) and a diffeomorphism on $S$, $\Psi: S \rightarrow S$ where $d\Psi\vert_{\bf q}:T_{{\bf q}}S \rightarrow T_{\Psi({\bf q})}S$ is the differential of $\Psi$ evaluated at ${\bf q} \in S$. For illustrative purposes, we'll consider local neighborhoods $N_{\bf p}, N_{\Psi({\bf p})} \subset S$ about an arbitrary point ${\bf p} \in S$ and $\Psi({\bf p}) \in S$.

If $\Psi$ is a local isometry, then $\forall {\bf q} \in N_{\bf p}$, $d\Psi\vert_{\bf q}$ can be associated with transformation in $\textrm{SO}(2)$ as $d\Psi$ preserves the inner product $$\langle {\bf v}_1, \ {\bf v}_2\rangle = \langle \ [d\Psi\vert_{\bf q}] {\bf v}_1, \ [d\Psi\vert_{\bf q}] {\bf v}_2 \rangle, $$ for all ${\bf v}_1, {\bf v}_2 \in T_{\bf q}S$.

Similarly, if $\Psi$ is locally conformal, there exists a differentiable function $\lambda^2:N_{\bf p}\rightarrow \mathbb{R}_{>0}$ such that for $\forall {\bf q} \in N_{\bf p}$, $$ \lambda^2({\bf q}) \langle {\bf v}_1, \ {\bf v}_2\rangle = \langle \ [d\Psi\vert_{\bf q}] {\bf v}_1, \ [d\Psi\vert_{\bf q}]{\bf v}_2 \rangle,$$for all ${\bf v}_1, {\bf v}_2 \in T_{\bf q}S$. It follows that for each ${\bf q} \in N_{\bf p}$, $d\Psi\vert_{\bf q}$ can be associated with an element of the Lie group $$\left\{ \alpha R \in \mathbb{R}^{2 \times 2} \ \mid \ R \in \textrm{SO}(2), \ \alpha \in \mathbb{R}_{>0} \right\}.$$

My question is as follows:

The above examples suggest that at least some types of diffeomorphisms on surfaces can be classified by associating the differential with a planar Lie group.

It seems that a natural next step would be to a define classes of diffeomorphisms whose differentials can be associated with $\textrm{SL}(2, \mathbb{R})$ and $\textrm{GL}(2, \mathbb{R})$, with the former possibly preserving something like local surface areas and the latter a notion of handedness.

I've looked around a bit but haven't yet been able to find a comprehensive treatment of diffeomorphisms that considers more "complex" types of transformations than isometries and conformal mappings, let alone anything that approaches the topic from more of a matrix Lie group perspective as I described above.

I'm hoping that someone might be able give me some information about any classes of mappings possibly associated with higher dimensional planar Lie groups (i.e. $\textrm{SL}(2, \mathbb{R}), \ \textrm{GL}(2, \mathbb{R})$ ). However, my knowledge of Riemannian/conformal geometry could charitably be described as limited, so it is likely that I'm unaware of well-known types of diffeomorphisms that fit the bill. In any case, pointing me towards a few resources that provide an in-depth treatment of more general classes of diffeomorphisms would be greatly appreciated.

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    $\begingroup$ Strictly speaking, $d\Psi|_{\bf{q}}$ is not an element of a Lie group, since it is a map between different vector spaces, and has no obvious choice of identity element. More accurately, the possible values of $d\Psi|_{\bf{q}}$ admit a free , faithful, and transitive left and right action by a single Lie group. It seems like the topic you're suggesting is closely related to Cartan geometry, which generalizes Riemannian geometry to other forms of "local structure" on manifolds from a very group-theoretic perspective. $\endgroup$
    – Kajelad
    Jul 8, 2020 at 22:08
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    $\begingroup$ You're close to reinventing frame bundles: A Riemannian manifold $(M, g)$ carries a canonical bundle whose fiber at $p$ is the space of all orthonormal bases of the inner product space $(T_p M, g_p)$; the right action of $O(g_p)$ realizes this bundle as a principle $O(n)$-bundle. Unwinding definitions shows that a diffeomorphism $\Psi : M \to M$ is in fact an isometry of $g$ iff pushes forward orthonormal bases to orthonormal bases. $\endgroup$ Jul 10, 2020 at 21:22
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    $\begingroup$ We can play the same game with other subgroups of $GL(n, \Bbb R)$, too, including the examples $GL_+(n, \Bbb R)$ (an orientable manifold), $SL(n, \Bbb R)$ (a manifold equipped with a volume form), $CSO(n, \Bbb R)$ (a conformal manifold), etc., $GL\left(\frac{n}{2}, \Bbb C\right)$ (an almost complex manifold). Cartan geometry, mentioned by Kajelad, is closely related to this notion, but it is more sophisticated and more powerful. $\endgroup$ Jul 10, 2020 at 21:28
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    $\begingroup$ @TravisWillse Thank you for your thoughtful and informative comments - this is exactly the kind of information I was looking for. Would you be willing to elaborate further in an answer, perhaps dealing with diffeomorphisms that push forward $\textrm{SL}(n, \mathbb{R})$? Regardless, do you know of a few introductory references the subject of frame bundles that deal with diffeomorphisms? $\endgroup$
    – tommym
    Jul 10, 2020 at 22:20
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    $\begingroup$ You're welcome, I'm glad you found it useful. I'm busy traveling for the next day or so, but I'll see if I can put together a useful answer for you soon. I learned the material from a draft of a (good but still-unreleased) textbook some years ago. One classic reference for the subject is: Kobayashi, Shoshichi (1972). Transformation Groups in Differential Geometry. Classics in Mathematics. Springer. $\endgroup$ Jul 10, 2020 at 23:30

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Much of this question is linear-algebraic in flavor:

For any point $p$ on a Riemannian manifold $(M, g)$ of dimension $n$, the tangent space $(T_p M, g_p)$ is an inner product space. Certain bases $(E_a)$ of $T_p M$ are well-adapted to the inner product, namely its orthonormal bases, those that satisfy $$g_p(E_a, E_a) = 1 \qquad \textrm{and} \qquad g_p(E_a, E_b) = 0$$ for all $a, b$, $a \neq b$. At least when $n > 1$, there are many choices of orthonormal basis of $(T_p M, g_p)$, and given any isometry $\phi : T_p M \to T_p M$ of $g_p$, we have $$g_p(E_a, E_b) = (\phi^* g_p)(E_a, E_b) = g_p(\phi(E_a), \phi(E_b)) ;$$ in particuar, if $(E_a)$ is an orthonormal basis, so is $(\phi(E_a))$. Thus, the group $O(g_p) \cong O(n, \Bbb R)$ acts transitively (and, in fact, freely) on the space (which below we'll denote $\mathcal F^O_p$) of orthonormal bases.

Back at the level of the Riemannian manifold $(M, g)$, unwinding definitions shows that the following are equivalent:

  • a diffeomorphism $\Phi : M \to M$ is an isometry;
  • for every $p \in M$ the differential $T_p \Phi$ is an isometry $(T_p M, g_p) \to (T_{\Phi(p)} M, g_{\Phi(p)})$ of inner product spaces;
  • for every $p \in M$ and any (equivalently, every) orthonormal basis $(E_a)$ of $T_p M$, $(T_p \Phi \cdot E_a)$ is an orthonormal basis of $T_{\Phi(p)} M$.

Put informally, a diffeomorphism is an isometry if it takes orthonormal bases to orthonormal bases.

This perspective suggests repackaging these ideas as follows:

For any smooth manifold $M$, the (tangent) frame bundle is the fiber bundle $\mathcal{F} \to M$ whose fiber $\mathcal F_p$ over $p$ consists of the bases of $T_p M$. The defining action of $GL(T_p M)$ on $T_p M$ takes bases to bases, so the induced action on the space $\mathcal F_p$ of bases realize $\mathcal F$ as a principal $GL(n)$-bundle over $M$. By definition, we may identify sections of this bundle with frames on $M$.

Likewise, for any Riemannian manifold $(M, g)$, the canonical orthonormal frame bundle is the bundle $\mathcal F^O \to M$ whose fiber $\mathcal F^O_p$ over $p$ consists of the orthonormal bases of $(T_p M, g_p)$, and by construction at each $p \in M$ the action of $GL(T_p M)$ restricts to the action of $O(g_p)$ described at the beginning of the answer. We may identify sections of $\mathcal F^O \to M$ with orthonormal frames.

In this language, any diffeomorphism $\Phi: M \to M$ induces a bundle isomorphism $\hat\Phi: \mathcal F \to \mathcal F$, and if $M$ is equipped with a Riemannian metric $g$, it is an isometry iff it maps $\mathcal F^O$ to itself. We call $g$---or, just as well, the frame bundle $\mathcal F^O$---an $O(n)$-structure.

Conversely, if we had started out with the $O(n)$-structure $\mathcal F^O$, we could have reconstructed the Riemannian metric $g$, and, as you suggest, we can ask what geometries can be realized as $G$-structures for other Lie subgroups $G \leq GL(n, \Bbb R)$, and for each what is the space of compatible bases/frames. For example:

  • The subgroup $GL_+(n, \Bbb R)$ of linear transformations of positive determinant corresponds to an orientation; the compatible bases are the positively oriented ones.
  • The subgroup $SL(n, \Bbb R)$ corresponds to a a volume form on $M$, i.e. a nonvanishing section $\Omega$ of $\bigwedge^n T^*M$; the compatible bases $(E_a)$ are those that span a parallelepiped of unit volume, i.e., for which $\Omega(E_1, \ldots, E_n) = 1$.
  • The subgroup $CO(n, \Bbb R)$ corresponds to a conformal structure $[g]$ on $M$; the compatible bases at $p$ are those that are an orthonormal for some metric $g_p$ in $[g_p]$.
  • The subgroup $GL(\frac{n}{2}, \Bbb C)$ corresponds to an almost complex structure, that is an endomorphism field $J : TM \to TM$ satisfying $J^2 = -1$ (essentially, this is an identification of each tangent space $T_p M$ with a complex vector space of dimension $\frac{n}{2}$). One natural choice for compatible bases are those of the form $(E_1, JE_1, \ldots, E_{n / 2}, J E_{n / 2})$ for some $(E_1, \ldots, E_{n / 2})$.

In all of the above examples, the definition of the geometric structure is essentially linear-algebraic, in that it can be characterized separately at each point (requiring only additionally that the structure vary smoothly from point to point). But many geometric structures are defined in part by differential conditions (often we call these integrability or nonintegrability conditions, depending on their character). For example, an almost complex structure $(M, J)$ defines a complex structure (i.e., a compatible maximal atlas of holomorphic charts) iff a certain tensor $N_J : \bigwedge^2 TM \to TM$---which in particular depends on the derivative of $J$---vanishes. In many cases, $G$-structures come equipped with a canonical connection (this is case for $G = O(n)$, i.e., for Riemannian manifolds, in which case the canonical connection is essentially the Levi-Civita connection), which can be used to study the differential behavior of the structure.

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    $\begingroup$ The answer already runs pretty long, so I don't want to include it there, but there's another instructive example opposite to the case $G = GL(n, \Bbb R)$: Take $G$ to be the trivial group. Then, there is exactly one adapted basis of each tangent space, that is the corresponding structure is nothing more than a frame itself; this structure is called an absolute parallelism on $M$, and it is equivalent to a bundle isomorphism $TM \to M \times \Bbb R^n$. $\endgroup$ Jul 12, 2020 at 22:22
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    $\begingroup$ "In many cases, 𝐺-structures come equipped with a canonical connection". The LC connection is currently the only example I know, what other canonical connections are there? $\endgroup$
    – Keshav
    Jul 13, 2020 at 14:44
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    $\begingroup$ One important example is the Chern connection associated to a Hermitian structure, which coincides with the LC connection only in the case that the Hermitian structure is Kähler. A recently studied example is sub-Riemannian structures; see arxiv.org/abs/1604.05392 . $\endgroup$ Jul 13, 2020 at 17:23
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    $\begingroup$ I've anyway removed the qualifier "$G$-principal" from that sentence in my answer to capture the idea better: Often one gets a canonical connection on some larger bundle; the most important example, at least historically, is Cartan's canonical connection associated to an ($n$-dimensional, $n \geq 3$) conformal structure, which can be viewed as what's now called a Cartan connection on a $P$-bundle $\mathcal G \to M$, and we call $(\mathcal G, \omega)$ a Cartan geometry of type $(SO(n + 1, 1), P)$, where $P$ is the stabilizer in $SO(n + 1, 1)$ of an isotropic ray/line. $\endgroup$ Jul 13, 2020 at 17:32
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    $\begingroup$ (It is not a coincidence that the Levi subgroup of $P$ is the group $CSO(n, \Bbb R)$ corresponding to $n$-dimensional conformal geometry.) Cartan's construction of a canonical connection on some "extended" bundle generalizes to many geometric structures, including (most of) the class of so-called parabolic geometries, which include among others projective, CR geometry, and various geometries of differential equations. NB that typically one does not get a canonical connection on the tangent bundle $TM$, and in a sense that can be made precise this is never the case for parabolic geometries. $\endgroup$ Jul 13, 2020 at 17:38

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