Cheeger Deformation

• Cheeger deformation is a canonical metric deformation built from an isometric compact Lie group action that preserves horizontal geometry while shortening orbit directions through the orbit tensor.
• As the deformation parameter enters the collapse regime, rescaled regular orbits converge in C^p to totally geodesic normal homogeneous spaces, with quantitative errors of order O(l²).
• The construction preserves nonnegative sectional curvature and can produce positive Ricci or scalar curvature, with applications to homogeneous spaces, Stiefel manifolds, exotic spheres, and proper Lie groupoid actions.

Cheeger deformation is a canonical metric deformation associated with an isometric action of a compact Lie group. It is constructed by taking a Riemannian product with a rescaled copy of the group and passing to a quotient Riemannian submersion. The deformation preserves the metric transverse to the group orbits while shortening orbit directions according to the orbit tensor. It was introduced in connection with nonnegative sectional curvature, but its structural effects persist without curvature assumptions: after suitable rescaling, orbit metrics converge on the regular part to normal homogeneous metrics and the orbit fibers become totally geodesic. Cheeger deformation also yields curvature-improvement and curvature-transfer mechanisms, with applications to positive Ricci curvature, almost nonnegative sectional curvature, homogeneous spaces, Stiefel manifolds, exotic spheres, and, more recently, proper Lie groupoid actions.

1. Construction and parameter conventions

Let $(M,g)$ be a complete Riemannian manifold and let a compact Lie group $G$ act effectively and isometrically on $M$. Fix a bi-invariant metric $Q$ on $\mathfrak g=\operatorname{Lie}(G)$. The classical construction equips $M\times G$ with the product metric

$g+\frac{1}{t}Q,\qquad t>0,$

and lets $G$ act by

$r\cdot(p,g)=(rp,gr^{-1}).$

The quotient is identified with $M$ through

$G$0

The unique metric $G$1 making $G$2 a Riemannian submersion is the Cheeger deformation of $G$3 at time $G$4 (Cavenaghi et al., 2018).

An equivalent convention uses a parameter $G$5, the product metric

$G$6

on $G$7, and the free Cheeger action

$G$8

The quotient map

$G$9

defines the metric $M$0 by requiring

$M$1

to be a Riemannian submersion (Searle et al., 2015). The parameters are inversely related in the usual descriptions: $M$2 corresponds to the collapse regime, whereas $M$3 corresponds to collapse in the $M$4-convention. In the latter convention,

$M$5

At $M$6, let $M$7 denote the isotropy subgroup and write

$M$8

as a $M$9-orthogonal decomposition. The orbit map identifies $Q$0 with the orbit tangent space: $Q$1 With

$Q$2

every tangent vector has a unique decomposition

$Q$3

The construction is geometric rather than an ad hoc tensor perturbation. The group factor supplies controlled directions corresponding to the action, and the quotient modifies precisely the metric components associated with those directions.

2. Orbit tensors and the deformed metric

The orbit tensor $Q$4 is defined by

$Q$5

It is positive definite and self-adjoint with respect to $Q$6. The tensor records the metric geometry of the orbit $Q$7 relative to the reference metric $Q$8.

For the deformed metric,

$Q$9

where

$\mathfrak g=\operatorname{Lie}(G)$0

If $\mathfrak g=\operatorname{Lie}(G)$1 has eigenvalue $\mathfrak g=\operatorname{Lie}(G)$2, the corresponding eigenvalue of $\mathfrak g=\operatorname{Lie}(G)$3 is

$\mathfrak g=\operatorname{Lie}(G)$4

Consequently,

$\mathfrak g=\operatorname{Lie}(G)$5

Thus orbit directions collapse at rate $\mathfrak g=\operatorname{Lie}(G)$6, while the metric on $\mathfrak g=\operatorname{Lie}(G)$7 remains unchanged.

The metric tensor $\mathfrak g=\operatorname{Lie}(G)$8 is defined by

$\mathfrak g=\operatorname{Lie}(G)$9

and satisfies

$M\times G$0

Equivalently, $M\times G$1 is the identity on horizontal vectors and contracts orbit vectors.

In the $M\times G$2-parameter convention, the corresponding orbit formula is

$M\times G$3

with equivalent formulations depending on whether the orbit tensor is regarded as an operator on $M\times G$4 or on $M\times G$5. The qualitative consequences are invariant under these conventions:

• horizontal directions are unchanged;
• orbit directions are shortened;
• $M\times G$6-invariance is preserved;
• the undeformed metric is recovered in the large-$M\times G$7 regime;
• after rescaling the orbit directions by $M\times G$8, the orbit metric converges to a normal homogeneous metric.

The Cheeger reparametrization provides an alternative description. If $M\times G$9 is the map associated with horizontal lifts, then

$g+\frac{1}{t}Q,\qquad t>0,$0

This identifies orbit tangent vectors with Lie-algebra directions perpendicular to the isotropy algebra.

3. Regularization of orbit geometry

The term “regularization” refers to geometric organization of the orbit directions, not analytic smoothing. If $g+\frac{1}{t}Q,\qquad t>0,$1 is smooth, every Cheeger-deformed metric is smooth. The regularizing effect is that arbitrary orbit metrics become asymptotically canonical.

Let $g+\frac{1}{t}Q,\qquad t>0,$2 denote the regular part, consisting of principal orbits, and let $g+\frac{1}{t}Q,\qquad t>0,$3 be $g+\frac{1}{t}Q,\qquad t>0,$4-invariant, open, and precompact. Define the rescaled metric

$g+\frac{1}{t}Q,\qquad t>0,$5

The horizontal distribution and horizontal metric are unchanged: $g+\frac{1}{t}Q,\qquad t>0,$6 for $g+\frac{1}{t}Q,\qquad t>0,$7.

For $g+\frac{1}{t}Q,\qquad t>0,$8, the orbit is

$g+\frac{1}{t}Q,\qquad t>0,$9

The fixed bi-invariant metric $G$0 induces a normal homogeneous metric $G$1 through

$G$2

If

$G$3

then the limiting metric $G$4 satisfies

$G$5

The map $G$6 is a Riemannian embedding and its image is totally geodesic (Searle et al., 2015).

The convergence is quantitative. On compact subsets of $G$7,

$G$8

where $G$9 is a uniformly $r\cdot(p,g)=(rp,gr^{-1}).$0-bounded symmetric tensor. Hence

$r\cdot(p,g)=(rp,gr^{-1}).$1

Theorem A establishes convergence in $r\cdot(p,g)=(rp,gr^{-1}).$2 for every fixed nonnegative integer $r\cdot(p,g)=(rp,gr^{-1}).$3. The estimates do not establish convergence in the full $r\cdot(p,g)=(rp,gr^{-1}).$4 topology because the constants controlling higher derivatives may depend on $r\cdot(p,g)=(rp,gr^{-1}).$5.

The orbit-space submersion

$r\cdot(p,g)=(rp,gr^{-1}).$6

has totally geodesic fibers in the limit. If $r\cdot(p,g)=(rp,gr^{-1}).$7 and $r\cdot(p,g)=(rp,gr^{-1}).$8 are the O’Neill $r\cdot(p,g)=(rp,gr^{-1}).$9-tensors, then

$M$0

Thus the second fundamental form of the orbits decays quadratically in the rescaled collapse parameter.

The result is confined to the regular part. Near singular or exceptional orbits, orbit dimensions and isotropy groups change, so the smooth orbit-tangent bundle description used in the estimates breaks down.

4. Curvature mechanisms

Cheeger deformation interacts with curvature through Riemannian-submersion geometry, orbit brackets, and the variation of the orbit foliation. After $M$1-reparametrization, for

$M$2

the fundamental sectional-curvature formula is

$M$3

where $M$4 and is nondecreasing in $M$5 (Cavenaghi et al., 2018). The bracket term records the noncommutativity of the group, while $M$6 measures interactions among orbit geometry, the horizontal distribution, and infinitesimal isotropy.

For a compact Lie group with bi-invariant metric, the group curvature is nonnegative, and the O’Neill contribution is also nonnegative. Consequently, nonnegative sectional curvature is preserved by Cheeger deformation. The deformation does not, however, create nonnegative sectional curvature from an arbitrary initial metric.

The Ricci curvature has a horizontal contribution and an orbit contribution. In the collapse limit, the orbit term approaches the Ricci curvature of the normal homogeneous orbit: $M$7 If the orbit $M$8 has finite fundamental group, there is a constant $M$9 such that

$G$00

This yields the positive-Ricci theorem: if $G$01 is compact, all $G$02-orbits have finite fundamental group, and

$G$03

then $G$04 has positive Ricci curvature for sufficiently large $G$05 (Sperança et al., 2017).

The singular-orbit analysis introduces an additional blow-up mechanism. At a singular point $G$06, let

$G$07

be the isotropy representation. For $G$08, define

$G$09

If

$G$10

vectors in $G$11 are called fake horizontal vectors with respect to $G$12. For such a vector $G$13,

$G$14

Therefore, nontrivial infinitesimal isotropy can generate curvature growing at least linearly in $G$15.

The exceptional case is a fixed axis: a nonzero $G$16 satisfying

$G$17

For a fixed axis,

$G$18

for every horizontal $G$19 and every $G$20, so

$G$21

Hence Cheeger deformation alone cannot overcome negative horizontal Ricci curvature in fixed-axis directions. This gives a negative answer to the question of whether Cheeger deformation always lifts positive Ricci curvature from a positively Ricci-curved quotient: conformal modification is essential in general (Cavenaghi et al., 2018).

Scalar curvature is more favorable. For non-Abelian $G$22, the scalar-curvature formula contains a bracket contribution that grows linearly in $G$23 on the regular part. Near singular orbits, the $G$24 estimate supplies further positivity. Consequently, if a compact manifold admits an effective isometric action by a compact group with non-Abelian Lie algebra, then sufficiently large Cheeger deformation produces positive scalar curvature. This gives a streamlined version of the Lawson–Yau result while preserving the full $G$25-symmetry.

5. Curvature criteria and geometric applications

For positive sectional curvature, the orbit geometry alone is insufficient. Suppose the normal homogeneous metrics on the regular orbits $G$26 have positive sectional curvature. A necessary and sufficient condition for sufficiently large Cheeger deformation to yield positive sectional curvature involves the quotient curvature, the Hessian of $G$27, the curvature form $G$28, and the covariant derivative $G$29. In the notation of the curvature criterion,

$G$30

for some $G$31. The condition expresses compatibility between quotient curvature, fiber variation, and the orbit connection (Cavenaghi et al., 2018).

For cohomogeneity-one manifolds, the quotient is an interval and cannot satisfy the quotient condition used in the positive-Ricci lifting theorem. Let $G$32 be the orbit tensor along a horizontal geodesic $G$33. If the principal orbits have finite fundamental group, sufficiently large Cheeger deformation has positive Ricci curvature precisely when there exists $G$34 with

$G$35

Choosing the orbit-tensor functions through disk-bundle geometry near the singular orbits recovers the Grove–Ziller theorem: a compact cohomogeneity-one manifold with two singular orbits and principal orbit of finite fundamental group admits an invariant metric with positive Ricci curvature.

Cheeger deformation also supports curvature transfer between equivariantly related manifolds. Consider a cross-diagram

$G$36

where $G$37 carries commuting group actions, $G$38 is a principal $G$39-bundle, and $G$40. A $G$41-invariant metric on $G$42 can be constructed by averaging, from natural metrics on pullbacks, or from an invariant connection and a Kaluza–Klein metric

$G$43

The common horizontal distribution projects isometrically to the horizontal distributions on both quotient manifolds.

Applying Cheeger deformation to $G$44 with respect to the principal action shrinks the principal vertical directions. If $G$45 is a positive lower bound for the eigenvalues of the principal orbit tensor, then

$G$46

Thus the negative vertical O’Neill contribution in the comparison of quotient sectional curvatures can be made arbitrarily small. The resulting estimates imply transfer of positive horizontal Ricci curvature between $G$47 and $G$48. If $G$49 is compact, $G$50 admits a $G$51-invariant metric with positive horizontal Ricci curvature if and only if $G$52 does (Sperança et al., 2017).

This mechanism applies to connected sums with exotic spheres, Kervaire manifolds, exotic projective-space analogues, sphere bundles over spheres, products of spheres, projective spaces, and homogeneous spaces. The resulting metrics include sequences satisfying

$G$53

The conclusion is positive Ricci curvature together with almost nonnegative sectional curvature, not generally nonnegative sectional curvature.

6. Homogeneous and Stiefel-space models

On a homogeneous space $G$54 with the left action of $G$55, there is one orbit type and the orbit space is a point. Cheeger deformation is then a deformation of homogeneous metrics, and the rescaled limiting metric is the normal homogeneous metric induced by $G$56 (Searle et al., 2015).

Stiefel manifolds provide an explicit model. Write

$G$57

Every tangent vector has the form

$G$58

The two-parameter family

$G$59

has

$G$60

The embedded metric is $G$61, while the canonical metric is $G$62.

Using

$G$63

these metrics are identified with Cheeger deformation metrics. The parameters satisfy

$G$64

Thus the canonical metric corresponds to $G$65, and the embedded metric corresponds to $G$66 (Nguyen, 2021).

The Ricci tensor is diagonal with respect to the $G$67- and $G$68-components: $G$69 The Einstein condition is

$G$70

For $G$71, the unique solution is

$G$72

For $G$73, both roots are positive: $G$74

The sectional-curvature numerator admits a sum-of-squares expression: $G$75 All terms are nonnegative for $G$76, so

$G$77

For $G$78 and $G$79, the exact nonnegative-curvature range is sharper: $G$80 For $G$81, positive curvature occurs for $G$82, nonnegative curvature at $G$83, and mixed curvature for $G$84. For $G$85, negative sectional curvature occurs for every $G$86; the complete curvature ranges in general are not established, although certain intervals are supported by explicit calculations and numerical evidence.

The Stiefel calculations arise from two equivalent methods: an embedded-manifold calculation using the Christoffel function and a homogeneous-space calculation using left-invariant Cheeger metrics on $G$87 followed by O’Neill’s formula. Their agreement gives a concrete instance of the general curvature calculus for Cheeger-deformed normal homogeneous spaces.

7. Extensions to Lie groupoids and limitations

Cheeger deformation has been extended from compact Lie group actions to proper Lie groupoid actions. Let

$G$88

be a proper Lie groupoid acting on a manifold $G$89 along $G$90. Its orbit decomposition produces a singular Riemannian foliation. A compatible $G$91-metric on the groupoid induces a $G$92-metric $G$93 on the arrow space, and a transversely invariant metric $G$94 can be chosen on $G$95 (Corro, 3 Feb 2025).

For $G$96, equip the fiber product $G$97 with

$G$98

The target map

$G$99

is a Riemannian submersion for a deformed metric $M$00 on $M$01.

The generalized orbit tensor

$M$02

is defined by

$M$03

Under the condition

$M$04

the deformed metric satisfies

$M$05

Thus only orbit directions are modified and the transverse metric is preserved. As $M$06,

$M$07

in $M$08. As $M$09, orbit directions collapse and

$M$10

in the Gromov–Hausdorff sense.

The generalized curvature formula combines O’Neill’s formula with the Gauss equation for

$M$11

It contains the original curvature, an arrow-space term multiplied by $M$12, an O’Neill term, and two second-fundamental-form corrections. Unlike the classical Lie-group case, the Gauss corrections have no automatic sign because the fiber product need not be totally geodesic. The arrow metric itself also need not have nonnegative sectional curvature. Consequently, nonnegative or positive sectional curvature is not automatically preserved for arbitrary proper Lie groupoid actions.

When the groupoid has one object, it is an ordinary compact Lie group, the fiber product is $M$13, the second fundamental form vanishes, and the generalized construction reduces exactly to classical Cheeger deformation. For general proper groupoids, the deformation applies to singular Riemannian foliations induced by groupoid actions, preserves the foliation and transverse distances, and collapses the leaves. A global deformation for arbitrary closed singular Riemannian foliations remains open because global holonomy groupoids need not exist and local groupoid deformations need not glue while retaining sectional-curvature control.