Exterior differential systems on compact three-manifolds and Cartan-Kähler theory

Question

Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:

$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$

together with the following condition:

$0 = [\sum_k^3 L_k e^k]\in H^1(M,\mathbb{R})$

where $(e^1,e^2,e^3)$ is a coframe on $M$ and $\Theta^i_{jk}\in C^{\infty}(M)$ are given functions on $M$ with $L_k := \Theta^{i_0}_{j_0 k}$ for some given $i_0$ and $j_0$ fixed. Every solution $(e^k)$ of the aforementioned system defines a natural metric as follows:

$h = \sum_k^3 e^k\otimes e^k$

I am interested in understanding the Riemannian metrics that can be defined in this way. This problem should be a particular case of an exterior differential system. However, I fail to see how to apply the theory correctly. First of all, I guess that we have to consider a larger manifold such that solutions to the system are integral submanifolds. I guess this should be the frame bundle of $M$. In any case, I don't fully understand how to apply the general theory of exterior differential systems, and which type of results this theory can provide in this problem. Also, it is not clear to me which results of the theory are purely local. Any help will be very welcome!

Thanks a lot!

Answer 1

In their comments to my first answer, the OP has clarified that they did not mean to regard the metric $h$ as a given, but, rather, an output of the problem of prescribing coframings by specifying their structure equations relative to certain given data. I don't want to erase my earlier answer, so I will now describe how to set up this different problem, which I would describe as follows:

One starts with a compact $3$-manifold $M$, a collection of smooth functions $f^i_{jk}$ on $M$ for $1\le i, j, k\le 3$, and a pair of integers $i_0$ and $j_0$ in the range $1\le i_0,j_0\le 3$. One seeks to find (or to determine whether there exists) a coframing $\omega = (\omega^i)$ on $M$ that satisfies

1. $\mathrm{d}\omega^i = \tfrac12 f^i_{jk}\,\omega^j\wedge\omega^k$, and

2. $f^{i_0}_{j_0k}\omega^k = \mathrm{d} a$ for some $a\in C^\infty(M)$.

Notes: First, the existence of a global coframing on $M$ implies that $M$ is orientable, so we might as well assume this from the outset. Also, it seems natural to assume that $f^i_{jk}=-f^i_{kj}$, but the OP does not specify this. For the purposes of Condition 1, this anti-symmetry assumption would be harmless, but Condition 2, as formulated, could have a different character without the anti-symmetry assumption.

Second, note that, if the three functions $f^{i_0}_{j_0k},\ 1\le k\le 3$ do not have any common zeros on $M$, then Condition 2 cannot ever be satisfied because the compactness of $M$ would ensure that the unknown function $a$ would have to have critical points, i.e., zeros of $\mathrm{d}a$. In practice, it seems better to weaken Condition 2 slightly by, instead, only requiring

2'. $\mathrm{d}\bigl(f^{i_0}_{j_0k}\,\omega^k \bigr)=0$,

since this is more directly expressed as a differential equation.

At any rate, Conditions 1 and 2/2', which are first order differential equations on $\omega$, clearly imply some zeroth order equations on $\omega$, that have to be taken into account: $$ \begin{aligned} 0 = \mathrm{d}(\mathrm{d}\omega^i) = \mathrm{d}\bigl(\tfrac12 f^i_{jk}\,\omega^j\wedge\omega^k\bigr)\bigr) &= \tfrac12\bigl(\mathrm{d} f^i_{jk}\wedge\omega^j\wedge\omega^k + Q^i(f)\,\omega^1\wedge\omega^2\wedge\omega^3\bigr)\\ 0 =\mathrm{d}(\mathrm{d}a) =\mathrm{d}\bigl(f^{i_0}_{j_0k}\,\omega^k\bigr) &= \mathrm{d}f^{i_0}_{j_0k}\wedge \omega^k + \tfrac12\,P_{jk}(f)\,\omega^j\wedge\omega^k \end{aligned} \tag 1 $$ where $Q^i(f)$ and $P_{jk}(f)$ are quadratic polynomials in the given functions $f=(f^i_{jk})$ that it will not be necessary to write out here. This is the vanishing of three $3$-forms and one $2$-form. When we regard $\omega=(\omega^i)$ as the usual tautological forms on the (trivial) principal $\mathrm{GL}(3,\mathbb{R})$-bundle $F\to M$ consisting of coframes of the tangent bundle of $M$, these equations will cut out a locus $B\subset F$, and solutions to the given problem will be coframings that are sections of $B\to M$. (Of course, not every section of $B$ will satisfy Conditions 1 and 2/2'.)

For generic choice of $f$, $B$ will be submanifold of $F$ of codimension 6, and the mapping $B\to M$ will be a submersion with fibers of dimension $3$, i.e., $B$ will be a $6$-manifold. Moreover, again, for generic choice of $f$, it can be shown that, at least locally in $B$, the three $1$-forms $\omega^i$ can be extended to a local coframing of $B$ by choosing three more $1$-forms $\pi_a$ ($1\le a\le 3$) for which there exist functions $A^{ia}_j$ on $B$ so that, pulled back to $B$ one has $$ \Theta^i = \mathrm{d}\omega^i-\tfrac12\,f^i_{jk}\,\omega^j\wedge\omega^k = A^{ia}_j\,\pi_a\wedge\omega^j, $$ and, in addition, for each $u\in B$, the only $3$-plane $E\subset T_uB$ that satisfies $E^*(\Theta^i)=0$ and $E^*(\omega^1\wedge\omega^2\wedge\omega^3)\not=0$ is given by $\pi_1=\pi_2=\pi_3=0$. Consequently, the only sections of $B$ that satisfy Condition 1 are the ones for which the forms $\pi_a$ vanish when pulled back to the graph in $B$. In other words, every coframing $\omega$ that satisfies Conditions 1 and 2/2' must be a section of $B$ that that is an integral manifold of the ideal generated by the $\pi_a$.

However, we must also take into account Condition 2', and one finds that, on $B$ one must have an expansion of the form $$ \Theta^4 = \mathrm{d}\bigl(f^{i_0}_{j_0k}\,\omega^k\bigr) = C^a_k\,\pi_a\wedge\omega^k + T_{jk}\,\omega^j\wedge\omega^k $$ for some functions $C^a_k$ and $T_{jk}=-T_{kj}$ on $B$. Since Condition 2' requires that $0 = E^*(\Theta^4)= E^*\bigl(T_{jk}\,\omega^j\wedge\omega^k\bigr)$ on any $3$-plane $E\subset T_uB$ that is tangent to the graph of a section $\omega$ satisfying Conditions 1 and 2/2', it follows that such sections must actually lie in the locus $B'\subset B$ defined by $T_{jk}=0$ ($1\le j < k\le 3$), which, generically is 3 independent equations on $B$, so that $B'\to M$ has discrete fibers over $M$.

Thus, in the generic situation, any coframing that satisfies Conditions 1 and 2/2' is found simply by differentiating the equations and imposing the zeroth order conditions that they imply.

Of course, it could happen that $B' = B = F$ in some cases, i.e., all of these zeroth order conditions are identities. This happens, for example, when the $f^i_{jk}$ are constants and $Q^i(f) = P_{jk}(f)= 0$ holds identically. In this case, of course, we know that all of the solutions are locally equivalent to the left-invariant coframing on a fixed Lie group $G$ of dimension $3$. (We also know that the compactness assumption implies further that there cannot be solutions on a compact manifold unless $f^{i_0}_{j_0k}=0$ for $1\le k\le 3$.)

There are many intermediate cases between these two extremes, depending on what identities and relations the functions $f = (f^i_{jk})$ satisfy on $M$. It would be very tedious to go through all the possibilities, so, unless one knows more about the functions $f$ on $M$, it is difficult to say much more in general about the problem and whether there is any need for any significant application of techniques from exterior differential systems.

Answer 2

Let me phrase the problem as I understand the given data and then describe how the 'theory of exterior differential systems' would be applied.

One starts with a compact Riemannian $3$-manifold $(M,h)$, a collection of smooth functions $f_{ijk}$ on $M$ for $1\le i, j, k\le 3$, and a pair of integers $i_0$ and $j_0$ in the range $1\le i_0,j_0\le 3$. One seeks to find (or to determine whether there exists) a coframing $\omega = (\omega_i)$ on $M$ that satisfies

1. $\omega$ is $h$-orthonormal, i.e., $h = {\omega_1}^2+{\omega_2}^2+{\omega_3}^2$,

2. $\mathrm{d}\omega_i = \tfrac12 f_{ijk}\,\omega_j\wedge\omega_k$ (summation on $j$ and $k$), and

3. $f_{i_0j_0k}\omega_k = \mathrm{d} a$ (summation on $k$) for some $a\in C^\infty(M)$.

Note: It seems natural to assume that $f_{ijk}=-f_{ikj}$, but the OP does not specify this. For the purposes of Condition 2, this anti-symmetry assumption would be harmless, but Condition 3, as formulated, could have a different character without the anti-symmetry assumption. I am going to impose the anti-symmetry condition for the following discussion, just to keep things simple. The more general case will be entirely similar.

One can actually go quite a ways without needing any of the special techniques of an analysis by the method of exterior differential systems (EDS). In particular, it seems quite likely that none of more advanced concepts of EDS, such as Cartan-Kähler theory, involutivity, etc. will actually be needed. The problem is so over-determined that elementary concepts involving differential forms will suffice.

The first thing to observe is that Condition 2 actually implies purely algebraic equations on the unknown coframing $\omega$: Applying the exterior derivative to both sides of the equations of Condition 2 yields $$ 0 = \mathrm{d}(\mathrm{d}\omega_i) = \mathrm{d}\bigl(\tfrac12 f_{ijk}\,\omega_j\wedge\omega_k\bigr) = \tfrac12\bigl(\mathrm{d} f_{ijk}\wedge\omega_j\wedge\omega_k + Q_i(f)\,\omega_1\wedge\omega_2\wedge\omega_3\bigr) \tag 1 $$ where $Q_i(f)$ is a homogeneous polynomial in the $f_{ijk}$ of degree $2$ for $1\le i\le 3$. Note that the above equations are three algebraic equations on the unknown $\omega$, which is a section of the $\mathrm{O}(3)$ bundle $F_h\to M$ of $h$-orthonormal coframes. It is easy to see (since $\mathrm{O}(3)$ has dimension $3$) that, for generic choice of the functions $f_{ijk}$, there will be only a finite number of sections of $F_h$ that satisfy these three conditions.

In this generic case, it would be easy to test each of the sections that satisfy these three equations by substituting them into the equations of Condition 2 to see which of those sections actually satisfy Condition 2. For each of the sections that do satisfy Condition 2, one can now check whether $\mathrm{d}(f_{i_0j_0k}\,\omega_k) = 0$. If this condition holds, one can then check whether $\bigl[f_{i_0j_0k}\,\omega_k\bigr] = 0\in H^1_{dR}(M,\mathbb{R})$. No methods of EDS are needed, and there's no particular help to be obtained from EDS to determine whether a cohomology class is trivial.

Of course, for special choices of $f = (f_{ijk})$, the three equations provided by (1) could be dependent, so that locus in $F_h$ where $(1)$ holds is, say, a smooth submanifold $B\subset F_h$ with fibers of $B\to M$ having positive dimension. It's conceivable that EDS could play a role in analyzing such cases, but I think it's unlikely that anything from EDS would ever actually be needed.

For example, it could happen that the equations (1) are trivial. It is easy to see that happens if and only if $\mathrm{d}f_{ijk}\equiv0$ and $Q_i(f)\equiv0$, i.e., the $f_{ijk}$ are constants and, moreover, they satisfy the so-called Jacobi equations. In particular, there exists a simply-connected $3$-dimensional Lie group $G$ endowed with a left-invariant coframing $\eta = (\eta_i)$ satisfying $\mathrm{d}\eta_i = \tfrac12 f_{ijk}\,\eta_j\wedge\eta_k\,$. In this case, there cannot be any coframing $\omega$ on $M$ satisfying Condition 2 unless the simply-connected cover $\tilde M\to M$ has a smooth map to $G$ that pulls back $\eta$ to be the pullback of $\omega$ to $\tilde M$. In particular, $(M,h)$ must be locally homogeneous.

To continue with the analysis in this case, switch viewpoints slightly and regard the $\omega_i$ as the canonical 'soldering' forms on the $\mathrm{O}(3)$-bundle $F_h\to M$ and let $\phi_{ij}=-\phi_{ji}$ be the $1$-forms on $F_h$ that satisfy Cartan's first structure equations. $\mathrm{d}\omega_i = -\phi_{ij}\wedge\omega_j$. Also, let $\alpha_{ij}=-\alpha_{ji}$ be the unique linear combinations of the $\omega_k$ such that $\alpha_{ij}\wedge\omega_j = -\tfrac12 f_{ijk}\,\omega_j\wedge\omega_k$. (By the Fundamental Lemma of Riemannian Geometry, such $\alpha_{ij}$ exist and are unique.) Now consider the $1$-forms $\theta_{ij} = \phi_{ij}-\alpha_{ij}$. Almost by definition, a section of $F_h$ that satisfies Condition 2 must be tangent to the $3$-plane field on $F_h$ defined by the equations $\theta_{ij}=0$, and any (local) section of $F_h$ that is tangent to this plane field is the graph of a (local) section of $F_h$, i.e., an $h$-orthonormal coframing, that satisfies Condition 2. Now we are reduced to an application of the Frobenius Theorem, which only depends on ODE, there is no need to appeal to anything like the Cartan-Kähler theory. (In actual practice, because of the necessary local homogeneity of $(M,h)$, one doesn't even need ODE or the Frobenius theorem; the problem reduces purely to algebra and elementary facts about Lie groups.

Of course, once one has found all of the coframings that satisfy Condition 2, it's very easy to check whether Condition 3 is satisfied. In fact, unless $f_{i_0j_0k}=0$ for all $k$, the $1$-form $\beta = f_{i_0j_0k}\,\omega_k$ will be a non-vanishing closed $1$-form on $M$, and hence it cannot be exact since, by hypothesis, $M$ is compact.