Let $M$ be a smooth Riemannian manifold of dimension $d$. Let $1 < k <d$ be an integer. Consider the exterior power bundle $\Lambda_{k}(T^*M)$.

Do closed and co-closed frames for $\Lambda_{k}(T^*M)$ always exist locally?

In other words, let $p \in M$. Does there exist a neighbourhood $U$ of $p$, which admits a frame of $k$-forms, all of which are closed and co-closed?

For $k=1$, the answer is positive, due to the existence of harmonic coordinates. Note that I do not require the frame to come "from coordinates" in any way; In Euclidean space, however, we can use coordinates of course; Take $dx^{i_1} \wedge \dots \wedge dx^{i_k}$, where $x_i$ are the standard coordinates.


Let us specialize to even dimension $d$, and let $k=\frac{d}{2}$.

Then, for a generic metric $g$, there are no coordinate systems where even one wedge is harmonic: $\delta(\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_n}) =0$. This implies harmonic frames, if exist generically, cannot be induced in general by coordinates. Moreover, if such a frame exists, then each member in it cannot be decomposable, since a closed decomposable form can always be expressed locally as the wedge of coordinate differentials.

In fact, such a harmonic frame $\omega^i$ must have the following property:

There are no non-zero decomposable elements, spanned by $\omega^i$ using constant coefficients. (i.e. there do not exist real numbers $a_i$ such that $\sum a_i \omega^i \neq 0$ is decomposable). Indeed, if such numbers existed, then $\sum a_i \omega^i$ would be decomposable and harmonic, which is generically impossible as mentioned above.

For a "generic" frame of $k$-forms, this property probably holds. (Here is an example for such "strongly non-decomposable frames").


1 Answer 1


$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\ep}{\epsilon}$

The answer is positive. Closed and co-closed frames always exist locally.

I will sketch here one proof: (For other approaches, see here and here). We want to prove that around every point $p \in \M$ there exist a local frame for $\bigwedge^k(T^*\M)$ whose elements are closed and co-closed.

For the Euclidean metric this is immediate: We have the standard (constant) frame $dx^I=dx^{i_1} \wedge \ldots dx^{i_k} $. Since every metric is locally close to being Euclidean on small neighbourhoods, the idea is to use an approximation argument:

Given a Riemannian metric $g$, we denote the space of $g$-harmonic forms of degree $k$ by $H^k_{g}$.

We view $H^k_{g}$, as a subspace of $\Omega^k(\M)$ which is "changing continuously" with the metric $g$. Suppose $g_{\ep} \to g_0$ in the $C^1$-norm where $g_0$ is the Euclidean metric; Then $H^k_{g_{\ep}} \to H^k_{g_0}$ in the following sense: there exist a family of bases of $H^k_{g_{\ep}}$, which converges to a basis of $H^k_{g_{0}} $ in $C^1$; this basis of $H^k_{g_{0}}$ forms a local frame for $\bigwedge^k(T^*\M)$. Since being a frame is an open condition, those bases for $H^k_{g_{\ep}}$ are local frames for sufficiently small $\ep$.

For the full details, see Appendix A in my paper here.

Some more details:

Even though the claim is local, and the approximation scheme is also inspired by a local phenomena, the implementation of the proof is based on a combination of local and global arguments. The reason is that on a closed manifold, being closed and co-closed is equivalent to being harmonic, and the dimension of the space of harmonic forms is a finite number which is a topological invariant of the manifold; it does not depend on the chosen metric.

Thus, given a family of metrics $g_{\ep} \to g_0$ on a closed manifold $\M$, we consider the behaviour of the finite-dimensional subspaces $H^k_{g_{\ep}}$ (all of the same dimension) as $\ep \to 0$.

That is, we look at the map $g \to H^k_{g}=\ker \Delta_g$. It turns out that this map is continuous in some appropriate sense; this relies on a certain "stability property of kernels of linear operators". It turns out that a crucial factor in the existence of such a stability phenomenon is the assumption that all the kernels have the same finite dimension. The convergence of kernels does not always hold when the dimensions are not equal or infinite.


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