Inflection Points of Projectively Flat Curves

Definitions and mathematical statement

Let $M$ be the Riemann surface in the source, regarded as a real surface, with its intended projectively flat connection. For a regular $C^2$ closed curve $\gamma:S^1\to M$ and a real torsion-free affine representative $\nabla$, a standard projective inflection condition is \[ \dot\gamma(t)\wedge\nabla_{\dot\gamma(t)}\dot\gamma(t)=0. \] Equivalently, the acceleration is tangent to the curve. This condition is unchanged by replacing $\nabla$ with a projectively equivalent torsion-free connection, since that replacement adds a tangential term to the acceleration. In a local real projective chart it is the ordinary inflection condition relative to straight geodesics.

Estimate the minimum possible number of inflection points of a closed curve in the stated geometry. For a specified admissible curve class $\mathcal C$, this would ask for a sharp lower bound on \[ \inf_{\gamma\in\mathcal C}|\{t\in S^1:\gamma\text{ has an inflection at }t\}|, \] with an appropriate multiplicity convention if required. The source does not specify $\mathcal C$, embeddedness, homotopy restrictions, genericity or isolated inflections. It also does not distinguish a real projective connection from a holomorphic projective structure. Thus the displayed real-connection definition explains one conventional interpretation, but the record does not supply a unique universal numerical bound or all hypotheses needed for one.

Short English statement

Find a sharp minimum number of inflections for closed curves on a surface with projectively flat geometry. The connection type, allowed curves, and counting convention are unspecified.

Sources

• Furuhata, Matsuzoe, and Urakawa, Open Problems in Affine Differential Geometry and Related Topics (1998), item 4(a), scanned PDF pages 1--2, https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; accessed 2026-07-29, manual OCR from page images recorded in the provenance. The record attributes the question to Takeshi Sasaki.
• The supplied summary mentions flex theorems and Plücker-type identities as partial context, without specifying publications or their hypotheses.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6000012$, source code catalogue entry, Inflection Points of Projectively Flat Curves. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.
• The supplied rights note says the J-STAGE article page marks this article CC BY 4.0.