Scalar Curvature Rigidity for Products of convex Hypersurfaces and even dimensional Manifolds

Abstract: We give a proof of scalar curvature rigidity in the spirit of Llarull and Goette-Semmelmann for products of strictly convex hypersurfaces in Euclidean space and nonnegatively curved spaces with non-vanishing Euler-characteristic. Our proof is based on the Fredholm family index theorem. This recovers corresponding results of Lockman-Zeidler where Clifford-linear (family) index theory is used.

• The paper proves that a degree-nonzero map with area-nonincreasing projections and scalar curvature at least that of the target must preserve scalar curvature when the target combines a nonnegatively curved spin manifold with nonzero Euler characteristic and strictly convex hypersurfaces.
• The authors use a geometrically constructed family of twisted Dirac operators and the classical Fredholm family index theorem, obtaining a nonzero index of degree(f)·2^k·χ(N) and converting curvature estimates into rigidity.
• Under the stronger condition scal > 2 Ric > 0, equality forces the map to be a local isometry and therefore a Riemannian covering, while extending the method to general odd-dimensional factors, flat tori, and low-regularity maps remains open.

Overview

This paper, by Frenck, Schick, and Schönlinner (2607.09472), establishes a scalar curvature rigidity theorem in the tradition of Llarull's theorem on the round sphere and the generalization by Goette–Semmelmann. The target manifolds considered are Riemannian products of a nonnegatively curved spin manifold with nonzero Euler characteristic and finitely many strictly convex hypersurfaces of Euclidean space. The proof is based on the classical Fredholm family index theorem for twisted Dirac operators, rather than the Clifford-linear index theory employed by Lockman–Zeidler (Lockman et al., 14 Jun 2026), whose results are recovered here by independent methods.

The main result is as follows. Let $S_i \subset \mathbb{R}^{n_i+1}$ be closed convex hypersurfaces with induced metrics $g_i$, where $n_i \geq 3$ is odd, and let $(N,g_N)$ be a closed spin manifold with nonnegative curvature operator and $\chi(N) \neq 0$. For any smooth map $f\colon (W,g) \to (M,g_M)$ from a closed connected spin manifold, where $M = N \times S_1 \times \dots \times S_k$ carries the product metric, satisfying:

• $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$,
• $\deg(f) \neq 0$,
• each composition $pr_N \circ f$ and $g_i$0 is area non-increasing,

the authors conclude that (i) $g_i$1, and (ii) if $g_i$2 holds on $g_i$3 and all $g_i$4, then $g_i$5 is a Riemannian covering.

Two remarks sharpen the statement. First, requiring only that the compositions with the projections be area non-increasing is genuinely weaker than requiring $g_i$6 itself to be area non-increasing, since products of area non-increasing maps need not be area non-increasing. Second, for a convex hypersurface $g_i$7 with $g_i$8, the condition $g_i$9 holds precisely when $n_i \geq 3$0 is strictly convex.

Relation to prior work

Llarull proved that the round metric on an even-dimensional sphere is scalar-extremal and rigid among metrics dominating it via degree-one maps. Goette and Semmelmann extended this to closed spin targets with positive curvature operator and nonzero Euler characteristic, using Fredholm index theory for twisted Dirac operators; there, the Euler characteristic condition enters through the K-theory cycle represented by the spinor bundle. For odd-dimensional spheres, where this cycle vanishes, Li–Su–Wang and Bär–Ziemke obtained proofs via spectral flow for strictly convex hypersurfaces. Lockman–Zeidler subsequently gave a uniform treatment of both parities using Clifford-linear family index theory, covering products of convex hypersurfaces with flat tori.

The present paper differs from Lockman–Zeidler in several respects. It uses the ordinary Fredholm index (and family index), possibly after taking the product with a round $n_i \geq 3$1, which should permit direct extension to comparison metrics and maps of low regularity via existing index theory for low-regularity Dirac operators. Its construction of the operator family is geometric: the family arises by pulling back from a higher-dimensional round sphere along an explicit smooth map $n_i \geq 3$2, rather than being built algebraically. Finally, the target may be any product of a manifold with nonnegative curvature operator and nonzero Euler characteristic with convex hypersurfaces, yielding a full extension of the Goette–Semmelmann rigidity result; however, torus factors, which Lockman–Zeidler handle with additional case distinctions, are deliberately omitted.

The index-theoretic argument

Assuming without loss of generality that $n_i \geq 3$3 is even (otherwise one takes the product with a round $n_i \geq 3$4-sphere), the authors construct a family of twisted Dirac operators parametrized by the torus $n_i \geq 3$5. The key geometric input is an explicit map $n_i \geq 3$6 built from a carefully chosen $n_i \geq 3$7-periodic function $n_i \geq 3$8: for $n_i \geq 3$9 the map sweeps latitudinal embeddings of $(N,g_N)$0 from south pole to north pole, and for $(N,g_N)$1 it traverses a longitudinal arc back. Each $(N,g_N)$2 has degree $(N,g_N)$3, and at integer multiples of $(N,g_N)$4 it restricts to an equatorial embedding. Combining these via the Gauss maps $(N,g_N)$5 yields a map

$(N,g_N)$6

with $(N,g_N)$7. Setting $(N,g_N)$8, the twist bundle $(N,g_N)$9 over $\chi(N) \neq 0$0 is defined as the pullback under $\chi(N) \neq 0$1 of the spinor bundle of $\chi(N) \neq 0$2 equipped with the metric $\chi(N) \neq 0$3.

Applying the family index theorem,

$\chi(N) \neq 0$4

the authors compute the total family index as

$\chi(N) \neq 0$5

using the standard fact that the graded spinor bundle represents the fundamental K-theory class multiplied by the Euler characteristic. Consequently, some member $\chi(N) \neq 0$6 of the family has nontrivial kernel — the crucial nonvanishing conclusion that replaces the odd-degree K-theory classes used in earlier work on odd dimensions.

The geometric estimate

The analytic core is a pointwise lower bound on the curvature endomorphism of the twist bundle, established through two lemmas. A linear-algebraic lemma states that for linear maps $\chi(N) \neq 0$7 and $\chi(N) \neq 0$8 with $\chi(N) \neq 0$9 and a bilinear form $f\colon (W,g) \to (M,g_M)$0, one has

$f\colon (W,g) \to (M,g_M)$1

where the $f\colon (W,g) \to (M,g_M)$2 are the singular values of $f\colon (W,g) \to (M,g_M)$3, with equality forcing $f\colon (W,g) \to (M,g_M)$4 for all $f\colon (W,g) \to (M,g_M)$5.

For each hypersurface factor, Gauss's theorema egregium gives $f\colon (W,g) \to (M,g_M)$6, so the contribution satisfies

$f\colon (W,g) \to (M,g_M)$7

For the $f\colon (W,g) \to (M,g_M)$8-factor, the nonnegative curvature operator hypothesis gives $f\colon (W,g) \to (M,g_M)$9. Summing over factors yields

$M = N \times S_1 \times \dots \times S_k$0

with two sharpness statements: the inequality is strict whenever some $M = N \times S_1 \times \dots \times S_k$1 is not a multiple of $M = N \times S_1 \times \dots \times S_k$2, and equality at $M = N \times S_1 \times \dots \times S_k$3 occurs exactly when $M = N \times S_1 \times \dots \times S_k$4 is an isometry.

Inserting this into the integrated Schrödinger–Lichnerowicz formula gives

$M = N \times S_1 \times \dots \times S_k$5

so $M = N \times S_1 \times \dots \times S_k$6 is invertible for all $M = N \times S_1 \times \dots \times S_k$7. Any element of the kernel must therefore live at $M = N \times S_1 \times \dots \times S_k$8 and be parallel; since the volume element acts constantly nonzero on such a spinor, the equality case forces $M = N \times S_1 \times \dots \times S_k$9 to be an isometry everywhere. As $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$0 is closed, $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$1 is a local isometry and hence a Riemannian covering onto its image; nonvanishing degree implies surjectivity, completing the proof.

The equality analysis adapts the singular-value argument of Goette–Semmelmann: assuming some singular value $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$2 of $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$3 leads to a contradiction because the strict inequality $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$4 guarantees enough positive singular values $\mathrm{scal}_g \geq (\mathrm{scal}_{g_M} \circ f)$5 that the relevant spans intersect trivially, contradicting the equality conditions in the linear-algebraic lemma.

Limitations and open questions

The paper concedes two significant restrictions. First, for even-dimensional sphere factors one may take any metric with nonnegative curvature operator, but for odd-dimensional factors the metric must currently be induced from a strictly convex codimension-one embedding into Euclidean space. The authors state plainly that they lack a method to represent nontrivial classes in odd K-theory by cycles coupled sufficiently to the Riemannian metric in full generality. Second, Gromov's question about lowering the regularity of the comparison map — motivated by the purely metric phrasing of the hypotheses — has been answered affirmatively elsewhere (by Cecchini–Hanke–Schick and related work, and independently by Lee–Tam), but extending the present method to such low-regularity settings remains open. Additionally, unlike Lockman–Zeidler, the theorem does not cover products with flat tori.

Conclusion

The paper provides an independent, geometrically constructed proof of scalar curvature rigidity for products of strictly convex hypersurfaces with nonnegatively curved Euler-characteristic targets, recovering the Clifford-linear results of Lockman–Zeidler via the classical Fredholm family index theorem. The explicit suspension-type family of maps into round spheres supplies the nontrivial index that parity obstructions otherwise preclude, and the resulting rigidity and covering conclusions extend the Goette–Semmelmann framework to mixed-dimension products. The principal open problems are the removal of the convex-embedding hypothesis on odd-dimensional factors and the extension of the method to low-regularity comparison data.