Manifolds modelled on flag manifolds — Question 2

Definitions and mathematical statement

The complex full flag manifold in ${\mathbb{C}}^3$ is \[ F_{12}=\{(L,P):L\subset P\subset{\mathbb{C}}^3,\ \dim_{\mathbb{C}} L=1,\ \dim_{\mathbb{C}} P=2\}. \] It has complex dimension three, with its natural complex structure $J$. For a smooth real three-manifold $M$, an immersion $f:M\to F_{12}$ is totally real if \[ D_xf(T_xM)\cap J D_xf(T_xM)=\{0\}\qquad(x\in M). \] Since the real dimension of $M$ equals the complex dimension of the target, these are maximally totally real tangent spaces.

The source asks for the homotopy classification of such immersions. For a fixed $M$, one natural formulation is to determine the path components of the space of totally real immersions: $f_0$ and $f_1$ are equivalent if there is a family $f_t$ through totally real immersions for $0\leq t\leq1$. This is a constrained regular-homotopy question, distinct from asking only which ordinary homotopy classes of maps contain a totally real immersion. The source does not choose between these readings or specify whether reparametrizations of $M$ are identified.

No assumptions on compactness, boundary, orientation, or topology of $M$ are supplied. A complete classification must say whether $M$ is fixed and include any relative boundary conditions. No CR, Lagrangian, or embedding condition is imposed by total reality, and none is added from the related-programme summary.

Short English statement

Classify totally real immersions of real three-manifolds in the complex three-dimensional full flag manifold. Specify whether equivalence means homotopy through such immersions or ordinary homotopy of maps, and fix the source-manifold class.

Sources

• Local UnsolvedMath registry, permanent ID 6800007, exact source code catalogue entry, "Manifolds modelled on flag manifolds---Question 2"; proposed by Elisha Falbel, Sorbonne Université, Paris, 2018. Registry status open, classification OPEN-TRIAGE.
• Morgan and Pansu, A List of Open Problems in Differential Geometry (2018), question environment 7, https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex. Supplied source-TeX extraction accessed July 29, 2026. The environment number is distinct from the question-2 title.
• The supplied summary places this in Falbel's flag-geometry/spherical-CR programme and says it found no verified full homotopy classification. It supplies no precise later reference or equivalence convention; related CR geometry is not made an extra hypothesis.
• Provenance flags current status and redistribution terms NEEDS_REVIEW; source TeX is publicly available on a coauthor's publication page.