Almost Nonnegative Curvature Operator (ANCO)

• ANCO is a curvature condition for closed smooth manifolds ensuring that, after normalizing by the squared diameter, the full spectrum of the curvature operator approaches nonnegativity.
• It bridges concepts from strict nonnegative curvature by allowing controlled lower spectral bounds, leading to topological constraints such as Betti number bounds and Euler characteristic inequalities.
• Techniques including principal bundle constructions and Ricci flow methods have produced diverse ANCO examples and stability results, highlighting its significance in geometric analysis.

Searching arXiv for recent and foundational papers on almost nonnegative curvature operator. Almost nonnegative curvature operator (ANCO) is a curvature condition on a closed smooth manifold requiring the full curvature operator to be asymptotically nonnegative after normalization by the square of the diameter. If $X$ is a closed smooth $n$-manifold, $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$ denotes the curvature operator of a Riemannian metric $g$, and $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$, $N=\tfrac{n(n-1)}2$, are its pointwise eigenvalues, then $X$ admits ANCO if there exists a sequence of metrics $\{g_i\}$ with $\diam(X,g_i)\le1$ and $\lambda_j(g_i)\ge-1/i$ for all $n$0. Equivalently, in the formulation of Herrmann–Sebastian–Tuschmann, for every $n$1 there is a metric $n$2 such that $n$3 (Herrmann et al., 2012, Cai, 25 Mar 2026). The subject has developed around three interconnected themes: construction of ANCO manifolds, comparison with manifolds carrying nonnegative curvature operator, and topological constraints such as Betti-number bounds, vanishing theorems, and Euler-characteristic inequalities (Herrmann et al., 2012, Teng et al., 29 Jul 2025, Cai, 25 Mar 2026).

1. Definition and equivalent formulations

For a closed Riemannian manifold $n$4, the curvature operator is the self-adjoint endomorphism

$n$5

characterized by

$n$6

Writing $n$7 for its smallest eigenvalue, ANCO means that for every $n$8 there is a metric $n$9 with

$\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$0

After rescaling to $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$1, this is equivalent to requiring all eigenvalues of $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$2 to lie in $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$3 (Herrmann et al., 2012).

A closely related formulation, used in later work, fixes a sequence of metrics $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$4 on a closed smooth $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$5-manifold $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$6 such that

$\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$7

This makes explicit that the full spectrum of the curvature operator is controlled from below, not merely the sectional or Ricci curvatures (Cai, 25 Mar 2026).

The literature also distinguishes almost flat curvature operator: if one has in addition

$\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$8

then the manifold has almost flat curvature operator, hence is infranil (Herrmann et al., 2012). In this sense ANCO is weaker than almost flatness, because it imposes only a lower spectral constraint.

2. Relation to nonnegative curvature operator

ANCO is modeled on the strictly nonnegative condition $\mathfrak R_g:\Lambda^2TX\to\Lambda^2TX$9, but the two notions are not equivalent. In the nonnegative case, the known structure theory is very rigid: if $g$0 is closed, simply connected, and $g$1, then $g$2 splits isometrically as a Riemannian product of round spheres $g$3, complex projective spaces $g$4 with the Fubini–Study metric, and compact irreducible Riemannian symmetric spaces of rank $g$5. In particular, in each fixed dimension there are only finitely many diffeomorphism types (Herrmann et al., 2012).

For ANCO manifolds, such a classification is not known. Nevertheless, some general topological bounds survive. A theorem of P. Bérard implies that if $g$6 admits ANCO, then for each $g$7,

$g$8

This is a global cohomological restriction derived by Bochner-type methods, and it shows that ANCO already constrains topology even without an actual nonnegative operator metric (Herrmann et al., 2012).

Later Ricci-flow results identify a noncollapsed regime in which “almost” nonnegativity of the curvature operator can be upgraded to genuine nonnegativity. Given $g$9, $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$0, and $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$1, there is $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$2 such that if $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$3 is closed with

$\lambda_1(g)\le\cdots\le\lambda_{N}(g)$4

then $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$5 admits a new metric $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$6 whose curvature operator lies strictly in $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$7; for $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$8 equal to the cone of nonnegative curvature operators, $\lambda_1(g)\le\cdots\le\lambda_{N}(g)$9 then carries a nonnegatively curved metric (Bamler et al., 2017). This does not classify ANCO in general, but it shows that under diameter and volume control the almost-nonnegative condition cannot wander far from the classical nonnegative theory.

3. Principal-bundle constructions and existence results

The first systematic constructions of ANCO manifolds beyond the nonnegative curvature-operator class were obtained through principal bundles. Let $N=\tfrac{n(n-1)}2$0 be a principal $N=\tfrac{n(n-1)}2$1-bundle with compact structure group $N=\tfrac{n(n-1)}2$2, connection $N=\tfrac{n(n-1)}2$3-form $N=\tfrac{n(n-1)}2$4, base metric $N=\tfrac{n(n-1)}2$5, and bi-invariant metric $N=\tfrac{n(n-1)}2$6 on $N=\tfrac{n(n-1)}2$7. For $N=\tfrac{n(n-1)}2$8, consider

$N=\tfrac{n(n-1)}2$9

Then $X$0 is a Riemannian submersion with totally geodesic fibres isometric to $X$1 (Herrmann et al., 2012).

The principal-bundle criterion is sharp. If $X$2 has $X$3, then $X$4 has ANCO as $X$5 if and only if

$X$6

where $X$7 is the curvature $X$8-form of the connection. If the base merely admits ANCO and $X$9 is abelian, then $\{g_i\}$0 admits ANCO. In particular, if $\{g_i\}$1 is a torus, any principal $\{g_i\}$2-bundle over an ANCO base admits ANCO (Herrmann et al., 2012).

These criteria produce explicit examples. For

$\{g_i\}$3

and coprime $\{g_i\}$4, let $\{g_i\}$5 be the principal $\{g_i\}$6-bundle with Euler class

$\{g_i\}$7

Each $\{g_i\}$8 admits ANCO. A Gysin-sequence calculation shows that infinitely many of the $\{g_i\}$9 are simply connected and have distinct homotopy types. Since only finitely many could support a metric with $\diam(X,g_i)\le1$0, this yields infinitely many closed simply connected $\diam(X,g_i)\le1$1-manifolds, and in each dimension $\diam(X,g_i)\le1$2 infinitely many closed simply connected $\diam(X,g_i)\le1$3-manifolds of pairwise distinct homotopy type, that admit ANCO but no metric with nonnegative curvature operator. There also exist closed, simply connected, homeomorphic $\diam(X,g_i)\le1$4-manifolds that admit ANCO but are not diffeomorphic (Herrmann et al., 2012).

The analytic mechanism behind the bundle construction comes from O’Neill’s formulas. In a suitable adapted basis, the curvature operator of $\diam(X,g_i)\le1$5 has block form

$\diam(X,g_i)\le1$6

and the leading mixed block is governed by the curvature form $\diam(X,g_i)\le1$7. If $\diam(X,g_i)\le1$8 has a nonzero component in $\diam(X,g_i)\le1$9, one obtains a negative eigenvalue of order $\lambda_j(g_i)\ge-1/i$0 as $\lambda_j(g_i)\ge-1/i$1, which obstructs ANCO; if $\lambda_j(g_i)\ge-1/i$2, the negative contributions go to zero as $\lambda_j(g_i)\ge-1/i$3 (Herrmann et al., 2012).

4. Ricci-flow formulations and almost-preservation

A second line of development studies almost nonnegative curvature conditions through Ricci flow. In Richard’s formulation, one fixes an $\lambda_j(g_i)\ge-1/i$4-invariant convex cone $\lambda_j(g_i)\ge-1/i$5 that is Ricci-flow invariant, contains the cone of nonnegative curvature operators, and is contained in the cone of operators with nonnegative Ricci curvature. A curvature operator $\lambda_j(g_i)\ge-1/i$6 is called $\lambda_j(g_i)\ge-1/i$7-almost $\lambda_j(g_i)\ge-1/i$8-nonnegative if

$\lambda_j(g_i)\ge-1/i$9

For the cone of nonnegative curvature operators, this is precisely the metric-level ANCO condition $n$00 (Richard, 2011).

The main preservation theorem in this framework states that for fixed dimension $n$01, constants $n$02, $n$03, and such a cone $n$04, there exist $n$05 and $n$06 with the following property: if $n$07 is a compact Ricci flow satisfying

$n$08

then for $n$09,

$n$10

This furnishes a short-time almost-preservation principle for lower curvature-operator bounds under Ricci flow (Richard, 2011).

Bamler–Cabezas-Rivas–Wilking proved a related theorem directly for curvature-operator eigenvalues. If $n$11 is complete, has bounded curvature, satisfies

$n$12

then the Ricci flow exists smoothly on $n$13 and for $n$14,

$n$15

where $n$16 and $n$17 depend only on $n$18 and $n$19. They also established analogous almost-preservation statements for other invariant cones, including $n$20-nonnegative curvature operator, PIC2, weakly PIC1, and nonnegative bisectional curvature in the Kähler case (Bamler et al., 2017).

These Ricci-flow results have structural consequences. In one direction, they yield Gromov–Hausdorff to Ricci-flow convergence statements and stability results showing that sufficiently small lower curvature-operator defects can be smoothed to genuine nonnegative curvature under additional injectivity-radius, diameter, or volume-ratio hypotheses (Richard, 2011, Bamler et al., 2017). In another direction, they provide smoothing procedures for certain singular limit spaces, including classes of Alexandrov spaces arising as limits of manifolds with lower curvature-operator bounds (Richard, 2011, Bamler et al., 2017).

5. Euler characteristic and other topological invariants

A central question in the subject, posed by Herrmann–Sebastian–Tuschmann, asks whether all compact ANCO manifolds have nonnegative Euler characteristic. The first partial affirmative answers came from index-theoretic methods based on a twisted Dirac operator associated with a closed $n$21-form $n$22. Defining

$n$23

one has

$n$24

Using a Bochner–Weitzenböck formula for $n$25, it was shown that if a compact $n$26-dimensional manifold admits ANCO and has $n$27, then its Euler number vanishes. In dimension four, the same approach gives $n$28, which resolves the Euler-characteristic question completely in the four-dimensional case (Teng et al., 29 Jul 2025).

Cai established a stronger general theorem under a two-sided spectral bound. If $n$29 is a closed smooth $n$30-manifold and $n$31 is fixed, then there exists $n$32 such that whenever a Riemannian metric $n$33 satisfies

$n$34

the Euler characteristic is nonnegative: $n$35 In particular, any ANCO manifold whose curvature operators also admit a uniform upper bound of this form has nonnegative Euler characteristic (Cai, 25 Mar 2026).

The proof combines geometric compactness and classical integral formulas. Two-sided spectral bounds imply a uniform bound on sectional curvature. When $n$36, the Chern–Gauss–Bonnet formula yields a uniform positive lower bound on volume. Cheeger–Gromov–Kasue compactness then gives a limit manifold $n$37 in $n$38, $n$39, and Weyl’s perturbation theorem shows that the curvature-operator eigenvalues converge almost everywhere to a nonnegative limit. Since a metric with nonnegative curvature operator satisfies $n$40 by a classical proposition of Bourguignon–Karcher and Kulkarni, a negative Euler characteristic is impossible (Cai, 25 Mar 2026).

Cai also proved a vanishing theorem under an ANCO-type lower bound and infinite fundamental group. If $n$41 is a closed smooth $n$42-manifold with infinite $n$43 and there exists a sequence of metrics $n$44 such that

$n$45

then

$n$46

and if $n$47 is spin, then $n$48 (Cai, 25 Mar 2026). The argument uses a linear-algebraic estimate of Petersen–Wink to obtain Bochner inequalities on differential forms, then applies $n$49-index vanishing on the universal cover; in the spin case, the hypothesis implies almost nonnegative Ricci curvature, and Chen–Ge–Han’s spin-index theorem gives the vanishing of $n$50 (Cai, 25 Mar 2026).

6. Classification status, comparisons, and open questions

The present theory exhibits a marked asymmetry between existence and classification. On the existence side, principal-bundle constructions provide abundant ANCO examples, including infinitely many simply connected manifolds that do not admit any metric with nonnegative curvature operator (Herrmann et al., 2012). On the rigidity side, Ricci-flow methods show that in the noncollapsed regime sufficiently small lower curvature-operator defects can force the existence of a genuinely nonnegative curvature-operator metric (Bamler et al., 2017). These results indicate that collapse is a major source of genuinely new ANCO phenomena.

Several foundational questions remain open. No full classification of ANCO manifolds is known. It is open whether in each dimension $n$51 there are ANCO spaces not admitting nonnegative curvature-operator metrics. It is also open whether every ANCO manifold virtually fibers with infranil or torus base and ANCO fibre, whether there are closed manifolds with almost nonnegative sectional curvature but no ANCO, whether exotic spheres carry ANCO, and whether ANCO manifolds are rationally elliptic (Herrmann et al., 2012).

The Euler-characteristic problem, originally listed among these open questions, has been substantially clarified but not in complete generality. The known results now show: $n$52 for compact even-dimensional ANCO manifolds with $n$53; $n$54 for compact ANCO $n$55-manifolds; $n$56 for closed even-dimensional ANCO manifolds under a uniform upper bound on the curvature operator; and $n$57 under an ANCO-type lower bound together with infinite fundamental group, with the $n$58-vanishing in the spin case (Teng et al., 29 Jul 2025, Cai, 25 Mar 2026). This suggests that the unresolved part of the Euler-characteristic question is concentrated in higher-dimensional, potentially collapsed, finite-$n$59 settings where no upper spectral control is assumed.