Let's say we have some surface embedded in a higher dimensional space. The space has curvature $K_1$. The surface has curvature $K_2$. Call the curvature on the surface $K_3$.
Is $K_3=K_1+K_2$?
What led me to guess this:
- In hyperbolic space, the surface of a horosphere is Euclidean: $K_1=-1,K_2=1,K_3=0$
- In Euclidean space, any surface has that curvature on its surface: $K_1=0,K_2=K_3$
- In any space, a plane has the same curvature on its surface as the space: $K_2=0,K_1=K_3$
These may just be special cases where $K_3=K_1+K_2$ happens to be true.
For $K_2<0$, embedding is not always possible. We may ignore the cases where such embedding is impossible.
I believe this is affirmed by Gauss' equation. It describes an embedded manifold's Riemann curvature, but it can be simplified to Gaussian curvature. In the link, $R$ corresponds to your $K_3$, and $R'$ to your $K_1$. (The equation is flipped and rearranged relative to yours; it says ${^-}K_1={^-}K_3+K_2$, or something like that.) The curvature of the surface relative to a Euclidean embedding space is
$$K_2 = \frac{\vec L_{uu}\cdot\vec L_{vv}-\vec L_{uv}\cdot\vec L_{vu}}{\lVert\frac{\partial\vec r}{\partial u}\wedge\frac{\partial\vec r}{\partial v}\rVert^2}$$
where $\vec r$ is the position vector, and $u$ and $v$ parametrize the surface.
The wedge product $\wedge$ has the same magnitude as the cross product, but it works in any number of dimensions; in 3D you can use $\frac{\partial\vec r}{\partial u}\times\frac{\partial\vec r}{\partial v}$ instead.
$\vec L$ is the Second Fundamental Form, which is basically the second derivative of the position vector, projected perpendicular to the surface. (The Christoffel symbols $\Gamma$ are its projection parallel to the surface.)
$$\frac{\partial^2\vec r}{\partial v\partial u} = \bigg(\frac{\partial^2\vec r}{\partial v\partial u}\bigg)^\parallel+\bigg(\frac{\partial^2\vec r}{\partial v\partial u}\bigg)^\perp$$ $$= \bigg(\Gamma_{uv}^u\frac{\partial\vec r}{\partial u}+\Gamma_{uv}^v\frac{\partial\vec r}{\partial v}\bigg)+\big(\vec L_{uv}\big)$$
These equations can be generalized to non-Euclidean embedding spaces, but you can't use a position vector.