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)
Notice that if $\theta_i=0$ for $i=1,\dots,n-1$, then your metric $g_{ij}$ fails to be Riemannian (it's not an inner product for $\theta_i=0$, because it's degenerate).
Hence, if your metric is not well-defined in some subset, you definitely cannot expect the inverse to be well-defined (in this case, it does not even exist!)
What you can do, however, is to change coordinates so that you cover the part of the manifold which is not covered by your first set of coordinates. The coordinates that induce your metric are the following: $$x_1={\cos{\theta_1}},$$
$$x_p={\cos{\theta_p}}{\prod_{m=1}^{p-1}}{\sin{\theta_{m}}},$$
$$x_n={\prod_{m=1}^{n-1}}{\sin{\theta_{m}}},$$
yet you can verify yourself that these are not one-to-one if one of the $\theta_i$ vanishes, so you can only have well-defined coordinates for the subset $$ \{(\theta_1,\dots,\theta_{n-1})\ |\ 0<\theta_i<\pi \textrm{ for }i\not=n-1,\ 0<\theta_{n-1}<2\pi\}. $$ If you seek another coordinate patch, you may for instance "flip" the sphere and build the coordinates in the exact same way, and then notice the relation between the two sets of coordinates (which will be the translation given by the "flip").