I am looking for a general formula for the Gaussian curvature of an $n$-sphere (the set of points in $R^{n+1}$ equidistant from the origin) of radius $r$.
From what I have read, there would be $n$ principal curvatures to consider, but since this space is so simple, I was hoping there would be a great deal of simplification. For a circle, the Gaussian curvature is $1/r$ and for a sphere it is $1/r^2$, but it seems too simple for it to be $1/r^n$ for $S^n$. However if it is indeed so, I would gladly welcome any sources, or just pointers in the right direction.
Best Answer
Your original hunch is correct. Gaussian curvature does make sense for a hypersurface, whereas for a general Riemannian manifold the curvature tensor and sectional curvature are appropriate. For a hypersurface $X$ in $\mathbb R^n$, the Gaussian curvature is the Jacobian of the Gauss map $\nu\colon X\to S^n$. For a sphere of radius $r$, we have $\nu(x) = x/r$, whose derivative is $1/r$ times the identity map. Thus, the determinant is indeed $1/r^n$.