# Example of a metric with negative scalar curvature everywhere on a sphere $S^n$

According to corollary 8.5.3 in [1], "There are no topological obstructions to negative Ricci or scalar curvature in dimensions at least $$3$$." More specifically, theorem 4.1 on page 8 in [2] says "Any smooth compact manifold $$M^n$$, $$n\geq 3$$ has a metric with negative scalar curvature."

This is surprising to me — so surprising that I find it hard to imagine. To cure my lack of imagination, I'll ask for an example...

For some $$n\geq 3$$, what is an example of a Riemannian metric defined everywhere on the sphere $$S^n$$ with everywhere-negative scalar curvature? I'm hoping for something written in the form $$ds^2 = \sum_{jk} g_{jk}(x) dx^j\,dx^k$$ with the coefficients $$g_{jk}(x)$$ given by explicit functions of the coordinates $$x$$, in some set of coordinate-patches that collectively cover the sphere.

References:

[1] Tuschmann and Wraith (2010), Moduli Spaces of Riemannian Metrics, https://link.springer.com/book/10.1007/978-3-0348-0948-1

[2] Notes by Li on Schoen (2017), "Topics in Scalar Curvature," https://geometrysummer.math.uconn.edu/wp-content/uploads/sites/2312/2018/06/Schoen_spring_2017__Topics_in_scalar_curvature.pdf

• The result is also mentioned in Wikipedia en.wikipedia.org/wiki/Scalar_curvature, weirdly under positive scalar curvature. According to the linked page on prescribed scalar curvature, by work of Kazdan and Warner, "If the dimension of M is three or greater, then any smooth function ƒ which takes on a negative value somewhere is the scalar curvature of some Riemannian metric" (M is assumed to be a smooth closed manifold).
– GFR
Aug 2, 2020 at 14:54

I used a CAS for the scalar curvature but I am sure it is not too hard to compute it by hand.

On $$S^3$$ take the Berger metric (also known as a squashed sphere), $$$$g=\eta _1^2+b^2\eta_2^2+c^2\eta_3^2$$$$ with $$\eta_i$$ left-invariant forms on $$SU(2)\simeq S^3$$.

A possible parametrisation is $$$$\begin{split} \eta _1 &= \sin \psi \, \mathrm{d} \theta - \cos \psi \sin \theta \, \mathrm{d} \phi ,\\ \eta _2 &=\cos \psi \, \mathrm{d} \theta + \sin \psi \sin \theta \, \mathrm{d} \phi,\\ \eta _3 &= \mathrm{d} \psi + \cos \theta \,\mathrm{d} \phi, \end{split}$$$$
$$b,c$$ are constants. The round metric on $$S^3$$ has $$b=c=1$$. We have $$\theta\in[0,\pi]$$, $$\phi\in[0,2\pi)$$, $$\psi\in[0,4\pi)$$.

The scalar curvature of this metric is $$$$s= - \frac{1}{2 b^4c^4} [b^8+(c^4-1)^2-2b^4(c^4+1)]$$$$ which clearly can be made negative by an opportune choice of $$b,c$$. For example for $$b=1$$, $$s=2-c^4/2$$ is negative for $$c^4>4$$.

• Thank you for the answer (+1)! This seems to be exactly what I wanted. For etiquette, I'll wait a while before accepting it, and in the meantime I'll spend some time checking it so I can reap some intuition. By the way, the expression you wrote for $s$ can also be written $$s = \frac{4B-(1+B-C)^2}{2BC}$$ with $B=b^4$ and $C=c^4$. Aug 3, 2020 at 1:16
• Just curious: what does CAS stand for? Computer-Aided-...? Aug 3, 2020 at 1:18
• Glad it was useful. It stands for Computer Algebra System!
– GFR
Aug 3, 2020 at 9:26