# 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 $$v = (v_1, v_2)$$

$$II_p(v,v) = (v_1,v_2) \cdot II_{p} \cdot (v_1,v_2)^T = v_1^2 \, r + v_2^2 \, r \sin^2 \theta \; ,$$

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