Consider the unit hypersphere in $\mathbb{R}^n$, i.e. with the Euclidean metric, using spherical coordinates.

The metric tensor is then:
$$ g_{11}=1 $$ $$ g_{ij}=\delta_{ij}\prod_{k=1}^{i-1}\sin^2(\theta_k) $$
where $\delta_{ij}$ is the Kronecker delta.
The inverse metric is then:
$$ g^{11}=1 $$ $$ g^{ij}=\delta_{ij}\prod_{k=1}^{i-1}\csc^2(\theta_k) $$
Notice that if *any* of $\theta_k=0$, then there will be a problem with $\csc(\theta_k)$. But $\theta_k=0$ seems to be a perfectly reasonable coordinate to be on.

Questions:

- (1)
**Why does the inverse metric fail to exist when $\theta_k=0$?** - (2)
**How do I fix this so that I can use the inverse metric***computationally*in cases where $\theta_k$ are allowed to freely vary (and vanish) as they can for $g$?

I'm hoping I'm making a silly mistake :)

(Note: this is true for $n=3$ too)