Riemannian metric on a model space and confusion with the notation $ds^2$.

Question

I'm having difficulties understanding some model Riemannian manifold. I am give $(\mathbb{R}^n, ds^2)$ where $ds^2 = dr^2 + f^2(r) d\theta^2$ and $d\theta^2$ is the Riemannian metric on the sphere $S^{n-1}$. I understand the Riemannian metric as a tensor field $g:M \to T^2(M)$ where $g_p:T_pM \times T_pM \to \mathbb{R}$ and the notation $ds^2, dr^2$ and $d\theta^2$ is confusing me. Is $ds^2$ defined to be the sum of two different metrics Riemannian metrics $dr^2$ and $d\theta^2$? How do I evaluate this $(ds^2)_p$ at some vectors $T_pM\times T_pM$? Do I have that $$(ds^2)_p(u,v)=(dr^2)_p(u,v) + f^2(r)d\theta^2(u,v)?$$ The matrix $g_{ij}$ corresponding to $ds^2$ should also be $$\begin{pmatrix}1&0\\ 0&f^2\left(r\right)\end{pmatrix}$$ right?

Answer 1

Yes, $ds^2$ is defined to be the sum of two metrics. Although, you have to be careful, because the notation you are using is hiding some subtleties. So, if we are on an $n-1$-sphere of radius $r$ then our metric is $r^2d \theta^2 = r^2 (d\phi^1 \otimes d \phi^1 + \sin^2(\phi^1) d\phi^2 \otimes d \phi^2 + ... + \sin^2(\phi^1) ... \sin^2(\phi^{n-2}) d\phi^{n-1} \otimes d\phi^{n-1})$, where the $\phi^1, \phi^2, ..., \phi^{n-1}$ are the angular coordinates on the $n-1$-sphere. Here lies the root of the issue (as far as I can tell). When you write down $ds^2$ or $d\theta^2$ in your notation, they are not elementary tensors. These objects are not computationally useful, they are merely symbolic and you need to use elementary tensors to actually calculate something. You can look up what an elementary tensor is, but in this problem $dr \otimes dr$ and the $d\phi^i \otimes d \phi^i$ are the elementary tensors in this coordinate system. So, if you were to expand out in terms of elementary tensors, you would be able to read off the components of your metric, which in this $n$-dimensional space should be an $n$ by $n$ diagonal matrix.

So, provided we have actually expressed our metric in elementary tensors, and we have vectors, preferably expressed in terms of the corresponding coordinate vectors, we can use the matrix above to actually calculate inner products. Otherwise, we can only write down something like $ds^2$ but not evaluate it (and get a number) in a literal sense.