Lie algebra valued 1-forms and pointed maps to homogeneous spaces

Question

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $(M,p_0)$ be a simply connected pointed smooth manifold. A $\mathfrak{g}$-valued 1-form $\omega$ on $M$ can be seen as a connection form on the trivial principal $G$-bundle $G\times M\to M$. Assume that this connection is flat. Then, parallel transport along a path $\gamma$ in $M$ from $p_0$ to $p$ determines an element in $G$, which actually depends only on the endpoint $p$ since we are assuming that the connection is flat and that $M$ is simply connected. Thus we get a pointed map $\Phi_\omega:(M,p_0)\to (G,e)$, where $e$ is the identity element of $G$. Now, the Lie group $G$ carries a natural flat $\mathfrak{g}$-connection on the trivial princiapl $G$-bundle $G\times G\to G$, namely the one given by the Maurer-Cartan 1-form $\xi_G$. Using $\Phi_\omega$ to pull-back $\xi_G$ on $M$ we get a $\mathfrak{g}$-valued 1-form $\omega$ on $M$ which, no surprise, is $\omega$ itself. So one has a natural bijection between $\{\omega\in \Omega^1(M,\mathfrak{g}) | d\omega+\frac{1}{2}[\omega,\omega]=0\}$ and pointed maps from $(M,p_0)$ to $(G,e)$.

(all this is well known; I'm recalling it only for set up)

If now $\omega$ is not flat but has holonomy group at $p_0$ given by the Lie subgroup $H$ of $G$, then we can verbatim repeat the above construction to get a pointed map $\Phi_\omega:(M,p_0)\to (G/H,[e])$. I therefore suspect by analogy that there should be a natural bijection

$ \{\omega\in \Omega^1(M,\mathfrak{g}) | \text{some condition}\} \leftrightarrow C^\infty((M,p_0),(G/H,[e])), $

but I've been so far unable to see whether this is actually true, nor to make explicit what "some condition" should be (it should be something related to the Ambrose-Singer holonomy theorem and to Narasimhan-Ramanan results on universal connections, but I've not been able to see this neatly, yet). I think that despite my unability to locate a precise statement for the above, this should be well known, so I hope you will be able to address me to a reference.

Answer 1

The question you are asking is a very basic one in the theory of what Élie Cartan called "the method of the moving frame" (in the original French, "la méthode du repère mobile"), so you should be looking that up. Cartan's basic goal was to understand maps of manifolds into homogeneous spaces, say, $f:M\to G/H$, by associating to each such $f$, in a canonical way, a 'lifting' $F:M\to G$ in such a way that the lifting of $\hat f = g\cdot f$ would be $\hat F = gF$ for all $g\in G$. If one could do such a thing, then one could tell whether two maps $f_1,f_2:M\to G/H$ differed by an action of $G$ by checking whether $F_1^*(\gamma) = F_2^*(\gamma)$, where $\gamma$ is the canonical $\frak{g}$-valued left-invariant $1$-form on $G$.

It turns out that it is not always possible to do this in a uniform way for all smooth maps $f:M\to G/H$ (even in the pointed category, which modifies the problem a little bit, but not by much). However, if one restricts attention to the maps satisfying some appropriate open, generic conditions, then there often is a canonical lifting $F$ for those $f$ belonging to this set of mappings, and it can be characterized exactly by requiring that the $1$-form $\omega_F = F^*(\gamma)$ satisfy some conditions. Working out these conditions in specific cases is what is known as the "method of the moving frame".

There's no point in trying to give an exposition of the theory here because it is covered in many texts and articles, but let me just give one specific example that should be very familiar, the Frenet frame for Euclidean space curves.

Here the group $G$ is the group of Euclidean motions (translations and rotations) of $\mathbb{E}^3$ and $H$ is the subgroup that fixes the origin $0\in\mathbb{E}^3$. The elements of $G$ can be thought of as quadruples $(x,e_1,e_2,e_3)$ where $x\in\mathbb{E}^3$ and $e_1,e_2,e_3$ are an orthonormal basis of $\mathbb{E}^3$.

When $f:\mathbb{R}\to\mathbb{E^3}$ is nondegenerate, i.e., $f'(t)\wedge f''(t)$ is nonvanishing, there is a canonical lifting $F:\mathbb{R}\to G$ given by $$ F(t) = \bigl(f(t),e_1(t),e_2(t),e_3(t)\bigr) $$ that is characterized by conditions on $\omega = F^*(\gamma)$ that are phrased as follows: First, $e_1\cdot df$ is a positive $1$-form while $e_2\cdot df = e_3\cdot df = 0$, and, second, $e_2\cdot de_1$ is a positive $1$-form while $e_3\cdot de_1 = 0$.

These conditions take the more familiar form $$ df = e_1(t)\ v(t)dt,\qquad de_1 = e_2(t)\ \kappa(t)v(t)dt, $$ for some positive functions $v$ and $\kappa$ on $\mathbb{R}$, and they imply $$ de_2 = -e_1(t)\ \kappa(t)v(t)dt + e_3(t)\ \tau(t)v(t)dt, \qquad de_3 = -e_2(t)\ \tau(t)v(t)dt, $$ for some third function $\tau$ on $\mathbb{R}$.

Conversely, any $F:\mathbb{R}\to G$ that satisfies the above conditions on $\omega = F^*(\gamma)$ is the canonical (Frenet) lift of a (unique) nondegenerate $f:\mathbb{R}\to\mathbb{E}^3$.

Without the nondegeneracy condition, the uniqueness fails. Just consider the case in which the image of $f$ is a straight line.

There are similar, but, of course, more elaborate, examples for other homogeneous spaces and higher dimensional $M$, but you should go look at the literature if you are interested in this.

Added in response to request in the comment: There are several excellent sources for the method of the moving frame. I'll just list (alphabetically by author) some of my favorites, which means the ones that I find most felicitous:

• Élie Cartan, "La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile", Paris: Gauthier-Villars, 1937. (His style takes some getting used to, and so many say that Cartan is unreadable, but, once you get used to the way he writes, there's nothing like Cartan for clarity and concision. I certainly have learned more from reading Cartan than from any other source.)

• Shiing-shen Chern, W. H. Chen, and K. S. Lam, "Lectures on Differential Geometry", Series on University Mathematics, World Scientific Publishing Company, 1999. (Chern learned from Cartan himself, and was a master at calculation using the method.)

• Jeanne Clelland, 1999 MSRI lectures on Lie groups and the method of moving frames, available at http://math.colorado.edu/~jnc/MSRI.html. (A nice, short elementary introduction.)

• Mark Green, "The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces", Duke Math. J. Volume 45, Number 4 (1978), 735-779. (Points out some of the subtleties in the 'method' and that it sometimes has to be supplemented with other techniques.)

• Phillip Griffiths, " "On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry", Duke Math. J. 41 (1974): 775–814. (Lots of good applications and calculations.)

• Thomas Ivey and J. M. Landsberg, "Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems", Graduate Studies in Mathematics, AMS, 2003. (Also contains related material on how to solve the various PDE problems that show up in the applications of moving frames.)

I think that this is enough to go on. I won't try to go into some modern aspects, such as the work of Peter Olver and his coworkers and students, who have had some success in turning Cartan's method into an algorithm under certain circumstances, or the more recent work of Boris Doubrov and his coworkers on applying Tanaka-type ideas to produce new approaches to the moving frame in certain cases.