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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?

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

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    $\begingroup$ Good god, how embarrassing. My concentration must haven gone down the drain after this long night of studying. Thank you, however for point that out. $\endgroup$
    – Octavius
    May 29, 2021 at 3:29

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