Minimal total absolute curvature for equiaffine immersions

Abstract: Koike (2001) defined the Lipschitz--Killing curvature and established a Chern--Lashof type inequality for equiaffine immersions of arbitrary codimensions. In this paper, we study the equality case. We prove that the total absolute curvature of an $n$-dimensional equiaffine immersion is equal to $2$ if and only if the image is a convex hypersurface embedded in an $(n+1)$-dimensional affine subspace.

• The paper proves that τ_S(f)=2 is equivalent to the immersion being a convex hypersurface in an (n+1)-dimensional affine subspace.
• It introduces a novel codimension reduction technique to preserve total absolute curvature without relying on traditional Euclidean metrics.
• The study extends classical Chern–Lashof results to the equiaffine context, linking Morse theory invariants with affine curvature properties.

Minimal Total Absolute Curvature for Equiaffine Immersions

Introduction and Context

The study establishes a Chern–Lashof-type characterization of convex hypersurfaces in affine differential geometry under the setting of equiaffine immersions. Whereas the classical Chern–Lashof theorem relates the minimal value of the total absolute curvature of an immersed compact manifold into Euclidean space with the property of being a convex hypersurface, this work extends such a result to the context of equiaffine immersions into an affine space of arbitrary codimension. The focus is on the geometric implications of total absolute curvature attaining the value two, an affine analog to the minimal Morse number for compact spheres.

Preliminaries and Existing Results

For an equiaffine immersion $$f: (M, \nabla, \theta) \rightarrow (\mathbb{A}^{n+r}, \tilde{\nabla}, \omega)$$ of an oriented compact $n$-manifold $M$ into an affine space, Koike introduced the Lipschitz–Killing curvature and the associated total absolute curvature $\tau_S(f)$ relative to a fixed unit ellipsoid $S$. The quantity $\tau_S(f)$, defined by integration over the transversal ellipsoid bundle $B$ of the absolute value of the Lipschitz–Killing curvature, generalizes the Chern–Lashof total absolute curvature to the equiaffine setting. Unlike the Euclidean case, $\tau_S(f)$ generally depends on the choice of $S$, but if it attains the minimal possible value (the minimum number of Morse critical points on $M$, which is $n$0 for spheres), this dependence becomes vacuous.

Koike proved that $n$1, where $n$2 are Betti numbers, and if $n$3, then $n$4 is homeomorphic to $n$5. However, the precise geometric characterization of the equality case $n$6 in the affine setting remained unresolved.

Main Theorem and Its Implications

The key contribution is the proof that for equiaffine immersions, the condition $n$7 is equivalent to the immersion image being a convex hypersurface embedded in an $n$8-dimensional affine subspace. This mirrors the classical Euclidean Chern–Lashof equality condition, but within the broader framework of affine differential geometry, free from reliance on any specific metric structure.

Theorem Statement

Given an equiaffine immersion $n$9 with $M$0 oriented, compact, and $M$1-dimensional, the following are equivalent:

• $M$2 for some (equivalently, for any) choice of unit ellipsoid $M$3,
• $M$4 is a convex hypersurface in an $M$5-dimensional affine subspace of $M$6.

This resolves the open geometric question from Koike's prior work by importing convexity as the sharp geometric criterion for minimal total absolute curvature in the equiaffine setting.

Technical Contributions

The proof strategy overcomes two core obstacles:

1. Reduction of Codimension: Unlike the metric (Euclidean) case, an affine subspace does not generally inherit a canonical equiaffine structure from the ambient space. A main technical device is the construction of an equiaffine structure on affine subspaces that preserves total absolute curvature during codimension reduction. Via iterative application, the ambient space may be reduced to $M$7-dimensional, with the immersion remaining equiaffine and the total absolute curvature unchanged.
2. Absence of Metric Structure: The arguments of Chern and Lashof rely fundamentally on Euclidean structure. Here, a novel approach is developed, relying on the behavior of height functions and the Gauss map associated to the immersion, without any dependence on metrics. The notion of a height function is transposed into affine geometry using the exterior algebra framework and the unit ellipsoid, substituting for the role of the unit sphere.

The equivalence is then established by:

• Showing that any immersion with $M$8 must lie in codimension one and admits supporting hyperplanes at each point, establishing convexity;
• Proving that any convex equiaffine hypersurface must have total absolute curvature $M$9, since all Morse height functions have exactly two critical points.

Examples and Generalizations

The paper presents explicit examples illustrating the theory:

• For the standard embedding of a sphere (ellipsoid) as an equiaffine immersion, the total absolute curvature is computed directly as $\tau_S(f)$0. All height functions (with respect to the unit ellipsoid) are shown to have exactly two critical points, aligning with the algebraic minimal Morse number.
• A nontrivial example is given of a convex hypersurface with degenerate affine fundamental form, illustrating that the convexity/minimal curvature equivalence does not require nondegeneracy of the induced affine metric. This points toward meaningful extensions of equiaffine geometry into the field of semi-definite and degenerate metrics, broadening possible objects of study.

Theoretical and Practical Implications

The results provide a purely global, intrinsic curvature criterion for convexity in affine differential geometry, valid in arbitrary codimension, and demonstrate that local strict convexity is not necessary for the minimal total absolute curvature characterization. This connects affine differential geometry more tightly to global Morse-theoretic invariants, independent of metric assumptions.

Further, the technical machinery developed for codimension reduction and the treatment of degenerate cases facilitates deeper connections with singularity theory, Kossowski metrics, and may find application in related fields, such as information geometry, where affine structures arise naturally.

There are also implications for the study of statistical manifolds, centro-affine geometry, and improper affine hyperspheres, as the affine-invariant methods for curvature and convexity extend beyond the Riemannian context.

Conclusion

This work provides a definitive answer to the geometric meaning of the minimal total absolute curvature for equiaffine immersions: it characterizes convex hypersurfaces in affine space, paralleling and generalizing the Euclidean Chern–Lashof theorem. The proof techniques avoid reliance on metric structures and remain valid for degenerate affine metrics, suggesting a broad scope for further generalizations in affine and information geometry. The results clarify foundational aspects of equiaffine immersion theory and open paths for future study of the geometry and topology of affine submanifolds with degenerate metrics and associated global invariants.