Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M,g)$ be a smooth Riemannian manifold, with curvature tensor $R$ and Levi-Civita connection $\nabla$. A scalar invariant of order at most one is a coordinate-independent algebraic expression formed from $R$, $\nabla R$ and the metric, for example a complete contraction of their tensor products. The source calls $(M,g)$ $1$-homogeneous when all such scalar invariants are constant on $M$. It is locally homogeneous if the local isometry pseudogroup acts transitively, equivalently any two sufficiently nearby points are related by a local isometry.

Statement recorded in the supplied source.

Is every $1$-homogeneous Riemannian manifold locally homogeneous? [S2]

Short English statement

Does constancy of all first-order scalar curvature invariants imply local homogeneity for positive-definite metrics?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100305 (AMR-020-0305). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Question 3.8. https://arxiv.org/abs/1804.03737.