For hypersurfaces of spheres, the Gauss map into the complex quadric is Lagrangian, and principal curvatures are the cotangent of local angle functions; the local converse is constructed explicitly.
Minimal Lagrangian submanifolds of the complex hyperquadric
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abstract
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all, respectively all but one, local angle functions coincide.
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math.DG 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
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Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres
For hypersurfaces of spheres, the Gauss map into the complex quadric is Lagrangian, and principal curvatures are the cotangent of local angle functions; the local converse is constructed explicitly.