Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar
\begin{equation}
K\_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}\_{\beta\mu\nu}R^{\beta}{}\_{\alpha\rho\sigma}\,,
\end{equation}
is a non-zero constant? $R^{\alpha}{}\_{\beta\mu\nu}$ is Riemann's curvature tensor and $\epsilon^{\mu\nu\rho\sigma}$ is the contravariant Levi-Civita tensor.


To be specific, I would be interested in finding a metric of such a spacetime.


In 3+1 dimensional theories of chiral (Weyl) fermions in a curved space-time, $K\_2$ is at source of gravitational anomaly -- a violation of charge conservation, where particles are produced at a rate proportional to $K\_2$ ([Jensen, K., Loganayagam, R. & Yarom, A. Thermodynamics, gravitational anomalies and cones](https://link.springer.com/article/10.1007/JHEP02(2013)088)). In chiral anomaly, a closely related phenomenon due to gauge fields, particle production rate is proportional to $\epsilon^{\mu\nu\rho\sigma}F\_{\mu\nu}F\_{\rho\sigma}=\mathbf{E}\cdot\mathbf{B}$, for which it's very easy to find a case where it's constant and non-zero.


My best effort so far was to find a power-series perturbation around a Minkowski spacetime in Cartesian coordinates $(t,x,y,z)$
\begin{equation}
g\_{\mu\nu}=\left(
\begin{array}{cccc}
 y^2-y z-z^2+1 & y^2-z^2 & 0 & 0 \\
 y^2-z^2 & y^2-y z-z^2-1 & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & -1 \\
\end{array}
\right)\,,
\end{equation}
for which
\begin{equation}
K\_2=\frac{32 \left(y z \left(2 y^2-y z-2 z^2\right)-1\right)}{\left(y z \left(2 y^2-y z-2 z^2\right)+1\right)^{5/2}}
= -32 + O\left((y^2+z^2)^2\right)\,.
\end{equation}
I can push the calculation to higher orders, but I'm not too hopeful in finding a closed-form expression this way.



Metric with a constant Chern–Pontryagin scalar
I found a solution:
\begin{equation}
g\_{\mu\nu} = \left(
\begin{array}{cccc}
 \frac{A}{(1-t z)^2 (x y-1)^2} & 0 & 0 & 0 \\
 0 & -\frac{A}{(1-t z)^2 (x y-1)^2} & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & -1 \\
\end{array}
\right)\,.
\end{equation}
The Chern-Pontryagin scalar is $K\_2 = -16/|A|$ for $xy>1,tz>1$.


**Update:** Under a conformal transformation $g'=\Omega g$, the Chern-Pontryagin scalar transforms as $K\_2' = K\_2/\Omega^2$, so for any metric with a non vanishing $K\_2$, one can take $\Omega = \sqrt{|K\_2|}$ and produce a metric, for which it's constant.


