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Classification conjecture for totally magnetic submanifolds of ellipsoids

Let a\mathbf a define the ellipsoid E(a)E(\mathbf a), with coordinate blocks of dimensions ℓ1,…,ℓk\ell_1,\ldots,\ell_k. For complex linear subspaces Vj⊆CℓjV_j\subseteq\mathbb C^{\ell_j}, consider the submanifolds

E(a)∩(V1⊕⋯⊕Vk).E(\mathbf a)\cap(V_1\oplus\cdots\oplus V_k).

Classification conjecture. Every closed totally magnetic submanifold of E(a)E(\mathbf a) is of the form

E(a)∩(V1⊕⋯⊕Vk).E(\mathbf a)\cap(V_1\oplus\cdots\oplus V_k).

The sphere case is cited as known, while the supplied text gives no resolution for general ellipsoids. This statement overlaps with the preceding conjecture but is stated with different notation and without the positive-dimension or connectedness restrictions, so it is retained separately pending verification of whether the source intends the claims to coincide.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Classification conjecture for totally magnetic submanifolds of ellipsoids

    Let AA be a positive Hermitian matrix with eigenspace decomposition corresponding to eigenvalue multiplicities ell1,…,ℓkell_1,\ldots,\ell_k, and let E(A)E(A) be the associated ellipsoid. For complex linear subspaces Vj⊆CℓjV_j\subseteq\mathbb C^{\ell_j}, consider the submanifolds

    E(A)∩(V1⊕⋯⊕Vk).E(A)\cap(V_1\oplus\cdots\oplus V_k).

    Classification conjecture. Every closed, connected totally magnetic submanifold of E(A)E(A) of positive dimension is of the form

    E(A)∩(V1⊕⋯⊕Vk).E(A)\cap(V_1\oplus\cdots\oplus V_k).

    For the sphere, the corresponding classification is known; the conjecture asks whether the block decomposition determined by the eigenspaces of AA gives all such submanifolds for general ellipsoids. The supplied text gives no resolution, so the conjecture remains open.

    source: Lina Deschamps, Levin Maier and Tom Stalljohann, “Topics in Magnetic Geometry: Interpolation, Intersections and Integrability”, arXiv:2604.13616 (2026).

Sources & referencesView supporting material

Primary source

Lina Deschamps, Levin Maier and Tom Stalljohann, “Topics in Magnetic Geometry: Interpolation, Intersections and Integrability”, arXiv:2604.13616 (2026).

ellipsoidsmagnetic-geometrysubmanifold-geometry

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