Curvature of $S^2$

Consider $S^2\subset\mathbb{R}^3$ parameterised by $\psi:S^2\to\mathbb{R}^3, (x_1, x_2)\mapsto (x_1,x_2,\sqrt{1-x_1^2-x_2^2})$.

We need to compute the sectional curvature of $S^2$ given this parameterisation. We are told that it suffices to compute this at the point $(0,0,1)$.

So we first obtain the $g_{ij}$ with a view to computing the Christoffel symbols. We found $g_{ij} = \begin{cases} 1+\frac{x_i^2}{1-x_1^2-x_2^2}, i=j\\ \frac{x_ix_j}{1-x_1^2-x_2^2}, i\neq j\\ \end{cases}$

At the point $(0,0,1)$, $g_{ij}=\delta_{ij}$, the Kronecker delta. Therefore at this point the Christoffel symbols and therefore the sectional curvature vanish.

My question is: how is it sufficient to compute the sectional curvature at this point if the Christoffel symbols vanish here? Does this not incorrectly imply that the sectional curvature of $S^2$ is 0?

Solution 1:

You already have an expression for the metric, you can compute its partial derivatives and evaluate them at $(0,0)$ in order to compute Christoffel symbols and derive the curvature. Hint: you should find $1$.

Your error is saying "since at $(0,0)$, $g_{ij} = \delta_{ij}$, then all partial derivatives vanish". This is basically an error that is equivalent to saying "all functions $f$ have zero derivative since at $x$, they are constant equal to $f(x)$."