# What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.

I know that on the 2-sphere we have $ds^2 = d\theta^2+\sin^2(\theta)d\phi^2$ (in spherical coordinates) but I'm not sure how this generalizes to $n$ dimensions.

Added note: If anything can be discovered only about the determinant of the tensor (when presented in matrix form), that would also be quite helpful.

I will define the metric of $$S^{n-1}$$ via pullback of the Euclidean metric on $${\mathbb{R}}^{n}$$.

To start with we take $$n$$-dimension Cartesian co-ordinates: $$(x_1,x_2......x_n)$$. The metric here is $$g_{ij }= \delta_{ij}$$, where $$δ$$ is the Kronecker delta.

We specify the surface patches of $$S^{n-1}$$ by the parametrization $$f$$: $$x_1=r{\cos{\phi_1}},$$

$$x_p=r{\cos{\phi_p}}{\Pi_{m=1}^{p-1}}{\sin{\phi_{m}}},$$

$$x_n=r{\prod_{m=1}^{n-1}}{\sin{\phi_{m}}},$$

Where $$r$$ is the radius of the hypersphere and the angles have the usual range.

We see that the pullback of the Euclidean metric $$g'_{ab} = (f^*g)_{ab}$$ is the metric tensor of the hypersphere. Its components are:

$$g'_{ab} = g_{ij} {\frac{\partial{x_i}}{\partial{\phi_a}}} {\frac{\partial{x_j}}{\partial{\phi_b}}} = {\frac{\partial{x_i}}{\partial{\phi_a}}}{\frac{\partial{x_i}}{\partial{\phi_b}}}$$

We get $$2$$ cases here:

i) $$a>b$$ or $$b>a$$, For these components one obtains a series of terms with alternating signs which vanishes, $$g'_{ab}=0$$ and thus all off-diagonal components of the tensor vanish.

ii) $$a=b$$,

$$g'_{11}=1$$

$$g'_{aa} ={r^2} \prod_{m=1}^{a-1} \sin^2{\phi_{m}}$$
where $$2\leq a\leq {n-1}$$

The determinant is very straightforward to calculate:

$$\det{(g'_{ab})} = {r^2} \prod_{m=1}^{n-1} g'_{mm}$$

Finally, we can write the metric of the hypersphere as:

$$g' = {r^2} \, d\phi_{1}\otimes d\phi_{1} + {r^2} \sum_{a=2}^{n-1} \left( \prod_{m=1}^{a-1} \sin^2{\phi_{m}} \right) d\phi_{a} \otimes d\phi_{a}$$

• Great, thanks for the step-by-step explanation. Although what you've written is fairly clear, you can make it clearer by using the TeX environment, with the $ delimiters, like instead of g'rϕk, write $g'_{r\phi_k}$, which comes out as$g'_{r\phi_k}$. – Jānis Lazovskis May 27 '13 at 11:39 • This is a very good answer and very useful too, as it is not that easy to find this piece of information on the net. Proper formatting would make it great. – Giuseppe Negro Oct 20 '14 at 9:12 • @GiuseppeNegro Done. Thanks for the feedback. – orange_soda Nov 10 '14 at 14:04 • It seems$g'_{11}=r^2$, no? – gamebm Nov 13 '18 at 14:17 • Can somebody explain more thoroughly why the off diagonal terms vanish? – Sigh_at_psi Apr 16 '19 at 11:15$\newcommand{\Reals}{\mathbf{R}}$For posterity: Fix$r > 0$, and let$S^{n}(r)$denote the sphere of radius$r$centered at the origin in$\Reals^{n+1}$. Stereographic projection from the north pole$N = (0, \dots, 0, 1)$on the unit sphere$S^{n} = S^{n}(1)$defines a diffeomorphism$\Pi_{N}:S^{n} \setminus \{N\} \to \Reals^{n}given in Cartesian coordinates by \begin{align*} \Pi_{N}(x_{1}, \dots, x_{n}, x_{n+1}) &= \frac{1}{1 - x_{n+1}}(x_{1}, \dots, x_{n}), \\ \Pi_{N}^{-1}(t_{1}, \dots, t_{n}) &= \frac{(2t_{1}, \dots, 2t_{n}, \|t\|^{2} - 1)}{\|t\|^{2} + 1}. \end{align*} In these coordinates, the induced (round) metric on the unit sphere is well-known (and easily checked) to be conformally-Euclidean: $$g(t) = \frac{4 (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t\|^{2} + 1)^{2}}.$$ Stereographic projection from the north pole(0, \dots, 0, r)$of$S^{n}(r)$is given by the scaled mapping$x \mapsto t = r\Pi_{N}(x/r)$, whose inverse is$t \mapsto x = r\Pi_{N}^{-1}(t/r)\$, i.e., \begin{align*} r\Pi_{N}(x_{1}/r, \dots, x_{n}/r, x_{n+1}/r) &= \frac{1}{r - x_{n+1}}(x_{1}, \dots, x_{n}), \\ r\Pi_{N}^{-1}(t_{1}/r, \dots, t_{n}/r) &= \frac{\bigl(2t_{1}, \dots, 2t_{n}, r(\|t/r\|^{2} - 1)\bigr)}{\|t/r\|^{2} + 1}. \end{align*} The induced metric in these coordinates is consequently $$r^{2} g(t/r) = \frac{4 (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t/r\|^{2} + 1)^{2}} = \frac{4r^{4} (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t\|^{2} + r^{2})^{2}}.$$