I am trying to show that the normal curvature of a sphere of radius $R$ is $\displaystyle\frac{1}{R}$.
I want to show this in a geometric intuitive way.
We know the gaussian curvature is given by the differential of the gaussian map at a given point.
In this case, since we are starting on a sphere of radius $R$ and projecting ourselves to a sphere of radius 1 (Gauss-Rodriguez map), yields:
$$\text{Gaussian Curvature of the sphere of radius $R$}=\text{det}\, dN_p=\displaystyle\frac{(dA)_{S^2}}{(dA)_S}=\displaystyle\frac{1}{R^2}$$
How can I conclude now the value of the normal curvature?
I would also appreciate if you could guide me through the differences between these curvatures. It seems like there are a lot of different curvatures and I am struggling on distinguishing them.
Thank you in advance for your time.