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

- 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.
- 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.
- 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.
- 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.)
- 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.
- $UTM$ 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 $T_qM$ ($p \neq q ∈ M$).