Intermediate curvature and splitting theorem

Abstract: In this paper, we prove several rigidity results for complete noncompact manifolds with nonnegative intermediate curvatures. We show that when either $3\leq n\leq 5$, $1\leq m\leq n-1$, or $6\leq n\leq 7$, $m\in {1,n-1,n-2}$, any manifold of the topological type $$M^{n-m}\times \mathbb{T}^{m-1}\times \mathbb{R}$$ with nonnegative $m$-intermediate curvature is isometrically covered by the canonical product $M\times \mathbb{R}<sup>m$. We also construct smooth metrics on $$M^{n-m}\times \mathbb{T}^{m-1}\times \mathbb{R}$$ with uniformly positive $m$-intermediate curvature for $6\leq n\leq 7$, $2\leq m\leq n-3$. This proves that the algebraic condition $m<sup>2-mn+m+n&gt;0$ from \cite{chenshuli_end} is sharp. The proof is based on a new recursion theorem for spectral intermediate curvatures and cylindrical splitting theorems. In particular, when $m=n-1$, this provides a new proof of some results by Chodosh--Li \cite{chodoshlisoapbubble} and Zhu \cite{zhu-splitting}. Moreover, the recursion theorem can be used to reprove the result of Brendle--Hirsch--Johne \cite{brendlegeroch'sconjecture}.

• The paper proves rigidity for complete noncompact manifolds with nonnegative m-intermediate curvature, showing they split as geometric products.
• It employs a spectral recursion theorem that reduces higher-dimensional curvature bounds to lower-dimensional slices using area-minimizing hypersurfaces.
• Sharp algebraic inequalities are validated by constructing explicit metrics, providing counterexamples to general splitting under relaxed conditions.

Rigidity and Nonexistence for Manifolds with Nonnegative Intermediate Curvature

Background and Motivation

The interplay between geometric curvature conditions and the global topology of manifolds is a central subject in Riemannian geometry. Classical results such as the Gauss-Bonnet theorem and the Bonnet-Myers theorem expose deep constraints linking curvature bounds to the topology and fundamental group properties. More recently, the structure and obstruction results for positive scalar curvature metrics—particularly the Geroch conjecture and its generalizations—have prompted the development of intermediate curvature notions that interpolate between Ricci and scalar curvature.

Intermediate ($m$-intermediate) curvature was formalized by Brendle-Hirsch-Johne, quantifying curvature via the sum of sectional curvatures across certain $m$-dimensional distributions. The $m$-intermediate curvature, denoted $C_m$, leads to a spectrum of rigidity and nonexistence results, encompassing both Ricci and scalar curvature as extreme cases. The paper "Intermediate curvature and splitting theorem" (2604.26529) advances this landscape by resolving rigidity for complete noncompact manifolds with nonnegative intermediate curvature and demonstrates the sharpness of algebraic dimension constraints underpinning these results.

Main Results and Claims

The central results pertain to manifolds of topological type $M^{n-m} \times \mathbb{T}^{m-1} \times \mathbb{R}$:

• Rigidity Theorem: For $3 \leq n \leq 5$, $1 \leq m \leq n-1$, or $6 \leq n \leq 7$, $m \in \{1, n-1, n-2\}$, any manifold of this type with nonnegative $m$-intermediate curvature is isometrically covered by the canonical product $m$0. Thus, such metrics must be geometric products with nonnegative Ricci components.
• Sharpness of Algebraic Inequalities: The authors construct smooth metrics with uniformly positive $m$1-intermediate curvature for $m$2, $m$3, violating the splitting phenomenon. This demonstrates that the algebraic constraint $m$4, previously conjectured to always suffice, is in fact sharp and cannot be relaxed.
• Recursion Theorem for Spectral Intermediate Curvature: The proof hinges on a spectral reduction theorem: a minimizer of a weighted area functional under spectral $m$5-intermediate curvature yields a lower-dimensional slice with inherited spectral $m$6-intermediate curvature. Iterating this reduction establishes the rigidity at the bottom slice.

Technical Approach

Spectral and Pointwise Intermediate Curvature

Intermediate curvature is defined for a collection of orthonormal vectors $m$7 in the tangent space via

$m$8

and the manifold is said to have nonnegative (resp. positive) $m$9-intermediate curvature if the minimum of $m$0 across all $m$1-frames at each point is nonnegative (resp. positive).

The spectral version incorporates a drift term: for a positive function $m$2,

$m$3

for some $m$4.

Cylindrical Splitting and Rigidity

The rigidity theorem relies on constructing a hierarchy of area-minimizing hypersurfaces (weighted by powers of $m$5), each yielding a lower-dimensional slice with spectral intermediate curvature. The bottom slice, essentially of the form $m$6, satisfies a spectral Ricci-type inequality. Using recent spectral splitting theorems (Antonelli-Xu, Catino et al.), these slices split as geometric products, provided the crucial algebraic inequality

$m$7

is satisfied, which is equivalent to $m$8. Failure of this inequality—demonstrated by explicit metric constructions—produces counterexamples to splitting, sharpness of admissible parameter ranges, and supports the necessity of these algebraic conditions.

The sequence of reductions generalizes the Cheeger-Gromoll splitting theorem, extending from Ricci to scalar (and intermediate) curvature regimes. The recursion theorem demonstrates the inheritance of spectral curvature bounds for minimizers, enabling inductive proofs of nonexistence theorems under intermediate curvature assumptions.

Construction of Counterexamples

For dimensions and intermediate parameters violating the key inequality, the authors construct metrics on $m$9 with uniformly positive $C_m$0-intermediate curvature, thus exhibiting non-splitting behavior. These constructions utilize warped products and ODE techniques, ensuring all sectional curvature inequalities are met, while avoiding product rigidity.

Numerical Results and Contradictory Claims

• Strong Claims: The algebraic condition $C_m$1 is proven sharp. Metrics exist outside the prescribed range with strictly positive $C_m$2-intermediate curvature but without splitting, refuting potential generalizations of previous rigidity theorems.
• Diameter Bounds: The authors derive sharp diameter estimates for weighted minimal slices, extending classical diameter bounds such as those of Bonnet-Myers and Shen-Ye to the intermediate curvature setting. The bounds are attained and shown to be asymptotically optimal by explicit manifold sequences.

Implications and Future Directions

Geometric Rigidity and Topology

These results clarify the precise relationship between intermediate curvature bounds and global geometric rigidity for large families of manifolds. The direct spectral and recursive arguments provide robust methods for translating curvature obstructions into topological constraints. Sharpness results indicate that further progress must materially account for the algebraic structure of dimension-curvature parameter spaces.

Extensions and Applications

The recursion theorem and splitting techniques generalize to spectral curvature operators beyond the intermediate setting (e.g., spectral scalar curvature, Ricci curvature), facilitating applications to noncompact manifolds, stability analyses, and diameter comparison theorems. The construction of counterexamples suggests fruitful directions for the study of exotic metrics and their geometric/topological properties.

The spectral splitting framework may be further expanded—recent advances such as the spectral splitting theorem for scalar curvature (Chai-Sun (Chai et al., 5 Apr 2026)) indicate ongoing generalizations, potentially encompassing other curvature operators, manifesting in rigidity and nonexistence phenomena across Riemannian geometry.

Open Questions

• Full characterization of minimizers in higher intermediate curvature settings and their recursive reduction properties.
• Extension of the spectral splitting theorem to broader curvature types and in higher codimensions.
• Further exploration of topological obstructions and their relation to curvature conditions outside the sharp algebraic thresholds.

Conclusion

The paper establishes substantial rigidity results for manifolds with nonnegative intermediate curvature, identifies precise algebraic sharpness bounds, and constructs explicit counterexamples outside admissible ranges. The spectral recursion theorem and the geometric splitting arguments form the technical core, linking curvature, homology, and manifold topology. These results both consolidate and extend the theory of curvature-topology obstructions, setting the stage for further developments in geometric analysis and spectral comparison theory.