Biorthogonal curvature

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 $S^2=\{x\in\mathbb R^3:\lVert x\rVert=1\}$ and $T^2=\mathbb R^2/\mathbb Z^2$. For a Riemannian metric $g$ on the four-dimensional product $M=S^2\times T^2$, a two-dimensional plane $\sigma\subset T_pM$ has a two-dimensional $g$-orthogonal complement $\sigma^\perp$. Write $K_g(\sigma)$ for sectional curvature computed from the Levi-Civita connection of $g$. Its biorthogonal curvature is \[K_{\mathrm{biort},g}(\sigma)=\tfrac12\left(K_g(\sigma)+K_g(\sigma^\perp)\right).\] Positive biorthogonal curvature means this quantity is strictly positive for every $p\in M$ and every two-plane $\sigma\subset T_pM$.

Statement recorded in the supplied source.

Does $S^2\times T^2$ admit a Riemannian metric with positive biorthogonal curvature? [S2]

Short English statement

Can the product of a sphere and a torus carry a Riemannian metric for which the average curvature of every pair of complementary orthogonal two-planes is positive?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6800004 (AMR-067-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexander Pigazzini, Positive Biorthogonal Curvature on $S^2\times T^2$ via Affine Connection, arXiv:2502.11914v11 (2025), Section 1, printed p. 1, and Section 1.2, "Problem (Bettiol, 2017)", p. 3. https://arxiv.org/abs/2502.11914v11.