Contracted Bianchi identity for infinite-dimensional Ricci curvature

Determine whether the contracted Bianchi identity holds for the Ricci curvature notions constructed from Zeitlin’s quantization, equivalently whether the associated Einstein tensor is divergence-free.

Background

In finite-dimensional Riemannian geometry, the contracted Bianchi identity implies that the Einstein tensor is divergence-free. The paper proposes Ricci curvature on infinite-dimensional fluid state spaces through finite-dimensional SU(N) approximations, but it does not establish whether this fundamental structural identity survives in the resulting infinite-dimensional framework. The authors identify this verification as a benchmark for assessing the geometric naturalness of competing definitions of infinite-dimensional Ricci curvature.

References

Finally, it would be insightful to examine whether the contracted Bianchi identity holds (equivalently, whether the Einstein tensor is divergence-free).

— Ricci curvature for fluid models on the torus via Zeitlin's quantization  (2609.01259 - Ishida et al., 1 Sep 2026) in Introduction, subsection “Future work”