Torus-symmetry thresholds for the Hopf conjecture under positive $k$-th intermediate Ricci curvature with $k\ge 3$

Hopf conjectured in the 1930s that every closed, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. The conclusion fails if sectional curvature is weakened to positive second intermediate Ricci curvature ($\mathrm{Ric}_2>0$): $S^3\times S^3$ and $S^7\times S^7$ carry such metrics with zero Euler characteristic. Kennard, Mouillé, and Nienhaus (2025) proved that under $\mathrm{Ric}_2>0$ a torus symmetry of rank at least ten restores the conclusion in dimensions divisible by four, and posed the natural extension: for each $k\ge 3$, does a dimension-independent torus-rank threshold $d(k)$ exist that forces positive Euler characteristic (or vanishing odd rational Betti numbers) under positive $k$-th intermediate Ricci curvature $\mathrm{Ric}_k>0$? The authors state that new ideas are required, since the connectedness principle underlying their proof loses one degree of connectivity for each unit increase of $k$. The question — with $k=3$ as the first open case — remains open.

Background

The Euler characteristic $\chi(M)$ of a closed manifold — the alternating sum of its Betti numbers, the dimensions of its rational cohomology groups — is constrained by curvature. Hopf conjectured in the 1930s that a closed, even-dimensional Riemannian manifold with positive sectional curvature has $\chi(M)>0$; this conjecture remains open in every dimension at least four and motivates a large part of global Riemannian geometry.

For an $n$-dimensional Riemannian manifold and an integer $k\in\{1,\dots,n-1\}$, the $k$-th intermediate Ricci curvature condition $\mathrm{Ric}_k>0$ requires that for every set of orthonormal tangent vectors $x,y_1,\dots,y_k$ the sum $\sum_{i=1}^k\sec(x,y_i)$ of sectional curvatures is positive, where $\sec(x,y)$ is the sectional curvature of the plane spanned by $x$ and $y$. Thus $\mathrm{Ric}_1>0$ is positive sectional curvature, $\mathrm{Ric}_{n-1}>0$ is positive Ricci curvature, and $\mathrm{Ric}_k>0$ implies $\mathrm{Ric}_\ell>0$ for all $\ell\ge k$; in particular $\mathrm{Ric}_k>0$ implies positive Ricci curvature, so Myers' theorem (finite fundamental group, hence vanishing first Betti number, in dimensions above two) is available for every $k$.

The Hopf conclusion fails under every relaxation of sectional curvature. Sha and Yang constructed even-dimensional connected sums of products of spheres with positive Ricci curvature and non-positive Euler characteristic, and Reiser and Wraith extended such constructions to $\mathrm{Ric}_k>0$ for $k$ somewhat larger than half the manifold dimension. Even at $\mathrm{Ric}_2>0$, the condition closest to sectional curvature, the conclusion fails: a metric constructed by Wilking on $S^3\times S^3$ has $\mathrm{Ric}_2>0$ and an isometric $T^3$-action with $\chi=0$, and DeVito, Domínguez-Vázquez, González-Álvaro, and Rodríguez-Vázquez built a $\mathrm{Ric}_2>0$ metric on $S^7\times S^7$ with a $T^4$-action. Since $\mathrm{Ric}_2>0$ implies $\mathrm{Ric}_3>0$, the failure of the Hopf conclusion already persists at $k=3$: the metrics above on $S^3\times S^3$ and $S^7\times S^7$ have $\mathrm{Ric}_3>0$ with $\chi=0$.

The Grove symmetry program asks whether large isometry groups restore curvature-driven topological rigidity. The symmetry rank of a Riemannian manifold is the maximal rank of a torus acting effectively and isometrically on it. Grove and Searle proved that positively curved $n$-manifolds have symmetry rank at most $\lfloor(n+1)/2\rfloor$, with a classification at equality. Wilking's Connectedness Lemma is the workhorse of the program: a map is $j$-connected if it induces isomorphisms on homotopy groups $\pi_i$ for $i<j$ and a surjection on $\pi_j$, and under positive sectional curvature the inclusion of a fixed-point component of an isometric action is highly connected; his Periodicity Lemma converts such connectivity into periodicity of the integral cohomology ring. Kennard's Four Periodicity Theorem deduces four-periodic rational cohomology in this situation, leading to Hopf-type theorems with symmetry: Kennard, Wiemeler, and Wilking proved that an isometric $T^5$-action on an even-dimensional positively curved manifold forces $\chi>0$, a bound improved to $T^4$ by Nienhaus.

For intermediate Ricci curvature, Mouillé extended the symmetry-rank bound, the isotropy rank lemma, and the connectedness principle: under $\mathrm{Ric}_k>0$, if a Lie group $G$ acting isometrically fixes a codimension-$d$ submanifold pointwise, the inclusion is $(n-2d+2-k+\delta(G))$-connected, where $\delta(G)$ is the dimension of the principal $G$-orbits — one degree of connectivity is lost for each unit increase of $k$. With these tools she proved that at large, dimension-dependent symmetry rank, $\mathrm{Ric}_2>0$ forces odd-dimensional simply connected manifolds to be spheres and even-dimensional ones (except in dimension six) to have positive Euler characteristic. Kennard and Mouillé classified maximal-symmetry-rank manifolds with $\mathrm{Ric}_2>0$ (spheres, complex projective spaces, and low-dimensional exceptions), and for $k\ge3$ obtained rigidity only under strong extra topological hypotheses such as $(k-1)$-connectedness or cohomology of a complex projective space up to degree $k+2$. No Euler-characteristic conclusion at any symmetry rank is known under $\mathrm{Ric}_k>0$ for any $k\ge3$.

The source paper proves the first dimension-independent threshold beyond sectional curvature: if a closed manifold $M$ with $\mathrm{Ric}_2>0$ and dimension divisible by four admits an isometric $T^8$-action, then every fixed-point component has vanishing odd rational cohomology, so $\chi(M)\ge0$; a $T^{10}$-action forces $\chi(M)>0$; and with connected isotropy groups, a $T^9$-action with a nonempty fixed-point set forces the rational cohomology to be four-periodic from degree $1$ to $n-1$, with corollaries that a $T^{10}$-action on a $4n$-manifold kills all odd rational cohomology and a $T^{11}$-action on a $(4n+1)$-manifold makes it a rational homology sphere. The key new tool is the Partial Four Periodicity Theorem, which lifts partial $\mathbb{Z}_p$-periodicity of the cohomology — periodicity holding only between degrees $1$ and $n-1$ — to rational four-periodicity, assuming $H^1(M;\mathbb{Z}_2)=H^1(M;\mathbb{Z}_3)=0$. The authors then pose this question, noting that for $k\ge3$ the connectivity loss in the connectedness principle starves the periodicity arguments, so new ideas would be required.

Problem Statement

For each integer $k\ge3$ — with $k=3$ the first open case — determine whether there exists an integer $d(k)$ such that every closed, even-dimensional Riemannian manifold $M^n$ with positive $k$-th intermediate Ricci curvature ($\mathrm{Ric}_k>0$) admitting an effective isometric action of a torus of rank at least $d(k)$ has positive Euler characteristic $\chi(M)>0$. The question is posed in this parameterized form by Kennard–Mouillé–Nienhaus; its scientific core is whether the Hopf conclusion, restored for $\mathrm{Ric}_2>0$ by dimension-independent torus symmetry in that paper, survives one further step away from sectional curvature. A complete answer breaks the $k=2$ barrier in one of two directions.

(1) Positive direction: for each $k\ge3$, exhibit an integer $d(k)$ and prove that every closed, even-dimensional manifold with $\mathrm{Ric}_k>0$ and an effective isometric torus action of rank at least $d(k)$ has positive Euler characteristic, or the stronger conclusion that all odd rational Betti numbers vanish. Since $\mathrm{Ric}_k>0$ implies $\mathrm{Ric}_\ell>0$ for $\ell\ge k$, a threshold theorem proved at a given $k$ transfers to every $3\le k'\le k$ but says nothing about larger $k'$; a complete positive answer therefore settles the question for every $k\ge3$.

(2) Negative direction: for some $k\ge3$, construct closed, even-dimensional manifolds with $\mathrm{Ric}_k>0$ and Euler characteristic at most zero admitting effective isometric torus actions of arbitrarily large rank, thereby proving that no finite threshold $d(k)$ exists for that $k$ (which settles the general parameterized question negatively).

Known anchors: for $k=2$, the threshold $d(2)=10$ works in dimensions divisible by four; for $k=3$, since $\mathrm{Ric}_2>0$ implies $\mathrm{Ric}_3>0$, the known zero-Euler-characteristic examples — the $T^3$-invariant metric on $S^3\times S^3$ and the $T^4$-invariant metrics on $S^7\times S^7$ — already satisfy $\mathrm{Ric}_3>0$, so any threshold must have $d(3)>4$ (in dimension four, where $\mathrm{Ric}_3$ coincides with the Ricci curvature, the product metric on $S^2\times S^2$ has $\mathrm{Ric}_3>0$ with a $T^2$-action, but $\chi=4>0$, so it is no obstruction); and symmetry-rank bounds cap the achievable rank in fixed dimension $n$ at $\lfloor(n+k)/2\rfloor-1$, so any positive $d(k)$ must exceed the ranks of all known zero-Euler-characteristic examples across all dimensions.

References

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