Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative.
For $w \in H$ consider the functional $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ where $\| \cdot \|$ denotes the standard norm in $\mathbb {R^3}$ and $| \cdot |$ the absolute value in $\mathbb{R}$.
I want to show that $\Omega := H(I;S^2) = \psi^{-1}(0)$ is a $C^\infty$-submanifold of codimension $1$ of $H$.
I think I'm at the point where I only need to make sure that the functional $\psi$ is smooth.
To compute the first derivative, I calculated the first variation which is $$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} \psi(w + \varepsilon \eta) = \int_0^1 \text{sgn}(\| w \|^2 - 1) \ 2 w\eta\ dt \ \ \ \ \ \ \text{ for all } \eta \in H$$
(which is a linear and bounded functional in $\eta$ and makes it the Frechét differential $D_w \psi$ at $w$ that is unequal $0$ making the implicit function theorem usable for the proof of the highlighted statement above.)
In turn $\Omega$ is at least a $C^1$-submanifold. Now calculating further derivatives / variations of $\psi$ to show smoothness, the second variation yields
$$\left.\frac{d}{d\varepsilon^2}\right|_{\varepsilon=0} \psi(w + \varepsilon \eta) = \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} \int_0^1 \text{sgn}(\| w \|^2 + 2\varepsilon w \eta + \varepsilon^2\|\eta \|^2 - 1) \ 2(w + \varepsilon\eta)\ dt \ \ \ \ \ \ $$
The derivative of the sgn-function is $0$ almost everywhere, continuing above terms with product rule yields (? not sure here)
$$ \int_0^1 \text{sgn}(\| w \|^2 - 1) \ 2 \eta\ dt $$
Is this correct?
