Difference between second fundamental form of $M$ in $\mathbb{R}^n$ and of $M$ as a boundary. How do I compute the second fundamental form and mean curvature of a boundary?
For example, if we consider the sphere $\mathbb{S}^n$ as a boundary of the hemisphere $\mathbb{S}^{n+1}$ with the spherical metric. Will the second fundamental form and mean curvature be different than that of $\mathbb{S}^n$ in $\mathbb{R}^{n+1}$?
I would like to see a worked example of these computation. I couldn't find it anywhere. Could show references?
 A: I am lazy to write up calculations, so let us take a look at the picture below. 

Your first example (on the left) is an equatorial embedding of $\mathbb{S}^n$ into the standard embedding of $\mathbb{S}^{n+1}$ into $\mathbb{R}^{n+2}$ (the round sphere). Try to convince yourself that the Gauss map is constant, and therefore the derivative of the unit normal vanishes identically. Such embeddings are called totally geodesic. You can learn more in the textbooks, or from this article (H.-B. Rademacher, Totally geodesic submanifold - definition).
In the second example, on the right, the Gauss map of the depicted embedding of $\mathbb{S}^n$ into $\mathbb{R}^{n+1}$ is not constant, so the derivative does not vanish. A hint to a calculation is given in this answer. This is an example of a totally umbilic embedding, by the way.
I hope that my answer will help you to recover accurate calculations.
Quiz. What is wrong with the right picture?
For the curious: I've just made this picture by hand in Inskcape with LaTeX formula extension.
