I'm trying to read a survey paper on the Willmore conjecture and I'm missing a lot of basic knowledge. In particular, let $u: \mathcal{M} \rightarrow S^3 \rightarrow \mathbb{R}^4$ be a smooth immersion of a compact orientable two dimensional surface into the standard 3-sphere, and let $\mathcal{M}$ take the metric induced by the ambient space. Writing the principal curvatures as $k_1$ and $k_2$, we have the Gaussian curvature given by $1+k_1k_2$.
I don't understand where the 1 in $K = 1+k_1k_2$ is coming from. The metric on $\mathbb{R}^4$ is the standard $g_{ij} = \delta_{ij}$, and restricting it to $S^3$ yields the round metric. $S^3$ is our ambient space, and restricting to $u$ gives us our induced metric.
I can think of two ways of showing this. The first would simply be to do everything out in local coordinates: for any $p \in \mathcal{M}$, we have a chart $\varphi: U \rightarrow V \subset \mathbb{R}^2$. Writing $u(p) = u(\varphi^{-1}(x^1,x^2)) = (y^1,y^2,y^3,y^4)$ and then projecting stereographically $\sigma: S^3 \rightarrow \mathbb{R}^3$, $\sigma(\vec{y}) = (\frac{y^1}{1-y^4},\dots,\frac{y^3}{1-y^4})$, we could recover $K$ via typical computations.
Alternatively, since $2K = \mathcal{R}$, the Ricci scalar, we could compute the curvature tensor and find it that way (unless there is some shortcut? I don't have much experience with these diffgeo objects).
I'm wondering if either these approaches would get me what I want, or if there is a naive reason for where the $1$ is coming from. Any help is appreciated. The relevant stuff is on p. 365 of the document, or 5th page from the start, section 2: the S^3 framework.