Normal curvature of sphere I am trying to calculate the normal curvature $\kappa$ of a sphere at point $p$ in direction $v$ which we defined as
$$ \kappa(p,v) = \frac{II_p(v,v)}{I_p(v,v)} \; , $$
where $I_p$ is the first fundamental form at $p$ and $II_p$ is the second fundamental form at $p$.
I parametrised the sphere in the usual way, which led me to the following matrix representation of the fundamental forms:
$$ I_p  =  \left( \begin{matrix} r^2 & 0 \\  0 & r^2 \sin^2 \theta \end{matrix} \right) \; , $$
$$ II_p =  \left( \begin{matrix} r & 0 \\  0 & r \sin^2 \theta \end{matrix} \right) \; . $$
(I checked on the internet and these matrices seem to be ok. Note that there is maybe a difference in sign, which arises from the choice of normal vector direction; inside pointing/outside pointing)
Now, I want to calculate $\kappa(p,v)$ for an arbitrary tangent vector $v = (v_1, v_2)$
which leads me to
$$ II_p(v,v) = (v_1,v_2) \cdot II_{p} \cdot (v_1,v_2)^T = v_1^2 \, r + v_2^2 \, r \sin^2 \theta \; , $$
and
$$ I_p(v,v) = (v_1,v_2) \cdot I_{p} \cdot (v_1,v_2)^T = v_1^2 \, r^2 + v_2^2 \, r^2 \sin^2 \theta \; . $$
I've read that the normal curvature of a sphere with radius $R$ is supposed to be $R^{-1}$, or at least constant everywhere on the sphere. My "ratio" $\kappa$ however is a function of $\theta$. What is my mistake here?

Pay attention
$$\frac{v_1^2r+v_2^2r\sin^2\theta}{v_1^2r^2+v_2^2r^2\sin^2\theta} = \frac{v_1^2r+v_2^2r\sin^2\theta}{r(v_1^2r+v_2^2r\sin^2\theta)} = \frac{1}{r}$$
