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I am considering the Sobolev space of unit tangent vector fields on a compact manifold: $Γ_{W^{1,2}}(M, UTM)$. I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$).

When $M$ is just a flat domain $Ω ⊆ ℝ^n$, those unit vector fields are functions into the sphere: $W^{1,2}(Ω, 𝕊^{n-1})$. For those there is the following theorem A:

Let $M$, $N$ be compact smooth manifolds, with $M$ simply connected. For $u ∈ W^{1,2}(M, N)$ there exists a sequence $(u^{(k)})_k$ with $u^{(k)} ∈ C^{∞}(M, N)$ so that $u^{(k)}$ converges weakly to $u$.

page 225 in Pakzad, M.R., Rivière, T.: Weak density of smooth maps for the Dirichlet energy between manifolds. Geom. Funct. Anal. 13(1), 223–257 (2003). doi:10.1007/s000390300006

I would like to proof the following version (Theorem B):

Let $M$ be compact simply connected smooth manifold. For $u ∈ Γ_{W^{1,2}}(M, UTM)$ there exists a sequence $(u^{(k)})_k$ with $u^{(k)} ∈ Γ_{C^{∞}}(M, UTM)$ so that $u^{(k)}$ converges weakly to $u$.

Can you point me to how to proof this, probably using Theorem A ?

Definitions

$W^{1,2}(M, N)$ is defined to be $\{u ∈ W^{1,2}(M, ℝ^N) ;\, u(p) ∈ N \text{ a.e. on } M\}$ where $N ⊆ ℝ^N$ is isometrically embeddded by the Nash theorem.

$Γ_{W^{1,2}}(M, TM)$ are measurable vector fields that are locally in charts weakly differentable and the norm $∫_M |u|_g + |∇u|_g \mathrm{d}\, V_g$ is finite with $∇$ being the Levi-Civita connection. Basis is Batu Güneysu. “Sobolev Spaces on Vector Bundles”. In: Dec. 2017, pp. 1–19. doi: 10.1007/978-3-319-68903-6_1.

$Γ_{W^{1,2}}(M, UTM) = \{u ∈ Γ_{W^{1,2}}(M, TM) ;\, |u|_g = 1 \text{ a.e. on } M\}$.

Actually I am interested in $UT^*M$ and $ℝ\mathbb{P}^{m-1}T^*M$ instead of $UTM$. But I assume any proof is pretty much the same for any of those.

What I tried and how it fails

  1. Reading the paper by Pakzad & Co, understanding and adapting the proof to my case. Problem: it goes far above my head. It probably would be a master thesis on its own.
  2. Using the density of smooth sections in $Γ_{W^{1,p}}(M, TM)$ (proven by Güneysu & Co). But this only gives me $|u|_g ≤ 1$, not equal to $1$ and Theorem A assumes $p = 2$ which somehow must be important.
  3. Look first at only a part $U$ of $M$ that admits an orthonormal frame. With this frame, we can write $u ∈ Γ_{W^{1,2}}(M, UTM)$ as $u \colon U → 𝕊^{m-1}$ ($m = \dim M$) and use Theorem A. But then it is unclear how to patch those different $U$'s together. We cannot simply use a partition of unity since $|u_p|_g$ is constant equal to 1.
  4. We embed $M$ in $ℝ^M$ (Nash embedding). Then $T_pM$ is embedded in $ℝ^M$ and $u_p ∈ 𝕊^{M-1}$. Then we use Theorem A. But then we do not know if $u^{(k)}_p ∈ T_pM$ is tangent to (the embedded) $M$. (Additionally I have not shown that $Γ_{W^{1,2}}(M, TM)$ defined via embedding is equivalent to my definition but I hoped to show that at some point anyway.)
  5. We embed $TM$ isometrically into some $ℝ^P$. Then it is not clear what $UTM$ looks like and if $u^{(k)}_p$ lies in $T_pM$. Also I am lacking a picture of the isometric embedding of $TM$ into $ℝ^P$ as I asked here. Answers to that are also highly appreciated.
  6. $UTM$ is a manifold itself. So we use Theorem A with $M=M$ and $N = UTM$. But then we don't know if $u_p^{(k)} ∈ T_pM$ or in some other tangent space $T_qM$ ($p \neq q ∈ M$).
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  • $\begingroup$ In case of interest: the master thesis which needs this is found here: hilsky.de/masterarbeitsrepo $\endgroup$
    – flukx
    Commented Mar 19, 2022 at 19:51
  • $\begingroup$ I just noticed that at least in dimension 2 there are not so many interesting cases. If there exists no smooth unit tangent field (e.g. Sphere), then the question is obviously moot. If it exists, you can take the orthogonal complement in each point and algebraic topology tells us that we can choose one of the two orientations. Then you have a global frame and my idea nr. 3 works by $U = M$.($TM$ is trivial.) But are there compact simply-connected 2-manifolds apart from sphere and a subset of $ℝ^2$? $\endgroup$
    – flukx
    Commented Mar 20, 2022 at 10:12

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