Affine Differential Geometry of Hypersurfaces

Affine differential geometry of hypersurfaces investigates the intrinsic and extrinsic properties of codimension-one submanifolds in affine space under volume-preserving linear transformations and translations. Central to this theory is the choice of an affine normal, a canonical transversal vector field that endows the hypersurface with an equiaffine structure. Equipped with the affine metric, a nondegenerate bilinear form induced by the second fundamental form relative to the affine normal, one studies curvature quantities such as the Pick form—a symmetric cubic tensor capturing deviation from quadric models—and the shape operator, which encodes how the affine normal varies along tangent directions. Of particular interest are affine spheres, hypersurfaces whose affine normals either intersect at a common point (proper) or remain parallel (improper). These arise as solutions to natural Monge–Ampère equations and feature prominently in convex geometry, integrable systems and mirror symmetry. Over the past decade, advances have addressed global classification, singularity theory and links to special Kähler and para-complex structures. Concrete applications range from computer vision algorithms for shape recognition to the study of special Lagrangian submanifolds in Calabi–Yau spaces.

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Technical terms

Affine normal: The distinguished transversal vector field on a hypersurface determined by the requirement that the induced volume form be preserved under parallel translation in affine space.

Affine metric: A nondegenerate bilinear form on the tangent bundle defined by the second fundamental form relative to the affine normal, serving as a pseudo-Riemannian metric on the hypersurface.

Pick form: A symmetric cubic tensor obtained by comparing the induced affine connection with the Levi-Civita connection of the affine metric, measuring deviation from quadratic models.

Affine sphere: A hypersurface whose affine normals are concurrent at a point (proper) or parallel (improper), equivalent to having constant affine mean curvature.

Shape operator: The endomorphism of the tangent bundle given by the negative derivative of the affine normal, encoding directional curvature information of the hypersurface.

References

1. On J~-tangent Affine Hyperspheres. Results in Mathematics (2020).
2. Affine Hypersurfaces of Arbitrary Signature with an Almost Symplectic Form. Results in Mathematics (2023).