Dropping the Gauss-map degree hypothesis from Chern–Lashof's Theorem 4: convexity of closed hypersurfaces with sign-constant Gauss–Kronecker curvature

Chern and Lashof's 1957 total-curvature paper proved (Theorem 4) that a closed, oriented hypersurface immersed in Euclidean space $\mathbb{R}^{n+1}$ is the boundary of a convex body if and only if its Gauss map $\nu$ has degree $\pm 1$ and its Gauss–Kronecker curvature $G$ (the determinant of the shape operator) does not change sign. The question is whether the degree hypothesis is redundant: must every closed, oriented, immersed hypersurface with $G$ of one sign be convex? The classical convexity lineage (Chern–Lashof's 1958 sequel, Hartman–Nirenberg, Sacksteder, do Carmo–Lima) assumes the strictly stronger condition of a semi-definite second fundamental form (all principal curvatures of one sign), which for $n=2$ — but not for $n\ge 3$ — is equivalent to $G$ of one sign; the case $n=2$ is therefore classical, and for odd $n\ge 3$ tube hypersurfaces are elementary non-convex examples with $G$ of one sign, so the case that remains open is even $n\ge 4$. There the question is equivalent to whether a tight immersion with Gauss-map degree $\chi(M)/2\ge 2$ and sign-constant $G$ can exist.

Background

Let $f\colon M^n\to\mathbb{R}^{n+1}$ be a $C^2$ immersion (a differentiable map with injective derivative everywhere) of a closed (compact, without boundary), connected, oriented $n$-manifold; the image is a hypersurface, possibly with self-intersections. Since $M$ is oriented one can choose a continuous unit normal field $\nu\colon M\to S^n$, the Gauss map. Its derivative defines the shape operator $A=-d\nu$, a self-adjoint endomorphism of each tangent space; the eigenvalues $\kappa_1,\dots,\kappa_n$ are the principal curvatures. The Gauss–Kronecker curvature is their product,

$G=\kappa_1\kappa_2\cdots\kappa_n=\det A,$

and equals the Jacobian determinant of the Gauss map, so the topological (Brouwer) degree of $\nu$ — the signed number of preimages of a regular value — satisfies $\deg\nu=(1/c_n)\int_M G\,dV$, where $c_n$ is the area of the unit sphere $S^n$. A convex body is a compact convex set with nonempty interior, and a hypersurface 'is convex' if it is embedded as the boundary of one.

Chern and Lashof (1957) founded the theory of total absolute curvature $\tau=(1/c_n)\int_M|G|\,dV$, proving $\tau\ge\beta(M)/2$, where $\beta(M)=\sum_i\dim H_i(M;\mathbb{Z}_2)$ is the sum of the mod-2 Betti numbers (the numbers of independent $i$-dimensional 'holes'); immersions attaining equality are called tight. Their Theorem 4 states for closed oriented hypersurfaces of $\mathbb{R}^{n+1}$ the equivalence of: (i) $\deg\nu=\pm 1$ together with $G\ge 0$ everywhere or $G\le 0$ everywhere; (ii) total curvature equal to that of a round sphere; (iii) $M$ is the boundary of a convex body. Where $G$ vanishes nowhere, (i) is automatic — an elliptic point (a point with definite second fundamental form) forces all principal curvatures to share a sign and $\nu$ to be one-to-one, which is Hadamard's theorem, the classical strict case. On the degenerate set $\{G=0\}$, however, the degree hypothesis genuinely restricts the class. Two further classical facts frame the question: Hopf's curvatura integra theorem (1926) states that for even $n$ the degree of the Gauss map is $\deg\nu=\chi(M)/2$, where $\chi$ is the Euler characteristic (for a modern elementary account of the underlying integral identity see Grinfeld 2026); and for odd $n$ the degree is not determined by the topology of $M$.

The neighboring, better-understood theory concerns infinitesimal convexity: the second fundamental form $II$ (the quadratic form $v\mapsto\langle Av,v\rangle$) is semi-definite, meaning all principal curvatures share a sign at each point, zeros allowed. By the Gauss equation (which computes the intrinsic sectional curvature of tangent 2-planes as $\kappa_i\kappa_j$ in flat ambient space), this is equivalent to nonnegative intrinsic sectional curvature. Hartman and Nirenberg (1959) and Sacksteder (1960) proved that a complete connected hypersurface with semi-definite $II$ and one elliptic point is the boundary of a convex body; do Carmo and Lima (1969) gave a Morse-theoretic proof of the compact case, and do Carmo and Warner (1970) treated spheres in spheres; the lineage is surveyed by de Lima (2022) and extended to negatively curved ambient spaces by Ghomi (2023). Semi-definiteness is strictly stronger than $G$ of one sign when $n\ge 3$ — for instance principal curvatures $(1,1,-1,-1)$ have $G=1>0$ but indefinite $II$ — whereas for $n=2$ the two conditions coincide (two numbers have nonnegative product exactly when they do not have strictly opposite signs). Modern work on the strict condition ($G$ nowhere zero) includes Zühlke (2020) and Ghomi (2002); a different curvature-scale family, q-convexity (the sum of the $q$ smallest principal curvatures is nonnegative), was analyzed by Colombo and Onti (2026) with topological conclusions for $q\le\lceil n/2\rceil$.

A standard source of examples is the tube construction: the set of points at fixed distance $r$ along the normals of a submanifold. The tube of radius $r<R$ around the round sphere $S^{n-1}(R)\subset\mathbb{R}^{n+1}$ is a closed embedded hypersurface diffeomorphic to $S^{n-1}\times S^1$ with principal curvatures $-\cos\varphi/(R+r\cos\varphi)$ of multiplicity $n-1$ and $-1/r$, so

$G=\frac{(-1)^n\cos^{\,n-1}\varphi}{r\,(R+r\cos\varphi)^{n-1}},$

which changes sign when $n$ is even but not when $n$ is odd; its Gauss map has degree $\pm 2$.

Problem Statement

Must every closed, connected, oriented $n$-manifold immersed in $\mathbb{R}^{n+1}$ whose Gauss–Kronecker curvature does not change sign be embedded as the boundary of a convex body, without assuming the degree of the Gauss map? Precisely: does the following hold for every even $n\ge 4$ — if $f\colon M^n\to\mathbb{R}^{n+1}$ is a $C^2$ immersion of a closed oriented manifold and the Gauss–Kronecker curvature $G=\det A$ — the determinant of the shape operator $A=-d\nu$, where $\nu\colon M\to S^n$ is the Gauss map, i.e. the continuous unit normal field determined by the orientation — satisfies $G\ge 0$ everywhere after possibly reversing $\nu$, then $\deg\nu=1$ and $f$ embeds $M$ as the boundary of a convex body? (The case $n=2$ is classical and excluded: $G\ge 0$ is then equivalent to a semi-definite second fundamental form, and convexity follows from the Hartman–Nirenberg, Sacksteder and do Carmo–Lima theory. The original formulation, an open question attached to Theorem 4 of Chern–Lashof 1957, asked this for $n\ge 2$.) In even dimensions $n\ge 4$ the question is equivalent, via Hopf's identity $\deg\nu=\chi(M)/2$ (with $\chi$ the Euler characteristic and $\beta$ the sum of the mod-2 Betti numbers) and the Chern–Lashof inequality $\tau\ge\beta(M)/2$, to: does there exist a tight immersion — one whose total absolute curvature $\tau=(1/c_n)\int_M|G|\,dV$ attains the Chern–Lashof lower bound $\beta(M)/2$ — of a closed oriented $2m$-manifold $M$ with $\chi(M)=\beta(M)\ge 4$ (all odd mod-2 Betti numbers zero) whose Gauss map has degree $\chi(M)/2\ge 2$ and whose Gauss–Kronecker curvature is of one sign? Odd dimensions $n\ge 3$ are excluded as well: the tube of radius $r<R$ around $S^{n-1}(R)\subset\mathbb{R}^{n+1}$ is a closed embedded non-convex hypersurface with $G$ of one sign and Gauss-map degree $\pm 2$, so the degree hypothesis cannot be dropped there.

References

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