Gromoll–Meyer Sphere

• The Gromoll–Meyer sphere is an exotic 7-sphere that is homeomorphic but not diffeomorphic to S⁷, constructed as a quotient of Sp(2) via commuting SU(2)-actions.
• It serves as a model for biquotient constructions, exhibiting nonnegative curvature and underpinning applications in transnormal, sub-Riemannian, and Lorentzian geometries.
• Its innovative quotient approach bridges classical differential geometry with modern extensions, influencing studies in curvature, geometric flows, and exotic manifold theory.

Searching arXiv for recent and foundational papers on the Gromoll–Meyer sphere, especially the provided 2024 paper and closely related geometric works. The Gromoll–Meyer sphere, usually denoted $\Sigma^7_{GM}$, is an exotic $7$-sphere: a smooth manifold homeomorphic but not diffeomorphic to the standard sphere $S^7$. It is realized as a quotient of $\mathrm{Sp}(2)$ by a free $\mathrm{Sp}(1)\cong S^3$-action, and it occupies a singular position in differential geometry as the first exotic sphere known to admit nonnegative sectional curvature and, in the biquotient description, as the only exotic sphere expressible as a biquotient of a compact Lie group (Nurfarisha et al., 2019). It has subsequently served as a model object for positive-curvature constructions, transnormal and sub-Riemannian structures, equivariant quotient constructions, and Lorentzian and Kaluza–Klein geometries (0805.0812).

1. Classical quotient construction and exoticity

A standard quaternionic description starts from

$$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$

with $a,b,c,d\in\mathbb H$. In this model, the first-row projection

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$

is a principal $\mathrm{SU}(2)$-bundle, where $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$ is the group of unit quaternions. The principal action $7$0 is

$7$1

while the Gromoll–Meyer action $7$2 is

$7$3

Its quotient is the Gromoll–Meyer exotic sphere, with quotient map

$7$4

so that

$7$5

The resulting pair $7$6 and $7$7 are repeatedly emphasized as “homeomorphic but not diffeomorphic manifolds” arising from the same total space by two commuting $7$8-actions (Cavenaghi et al., 2024).

A closely related formulation treats $7$9 as a biquotient. In that language, the quotient is written as $S^7$0, with the $S^7$1-action acting on the left by $S^7$2 and on the right by $S^7$3 (Nurfarisha et al., 2019). This biquotient presentation is not merely notational: it underlies the metric, curvature, and equivariant constructions that dominate the later literature.

2. Commuting actions, cross-diagrams, and pullback models

The quotient construction is most naturally organized by a commuting-action diagram. In the language of $S^7$4-diagrams, if $S^7$5 is a principal bundle with principal action $S^7$6, and there is another free action $S^7$7 of the same compact connected Lie group $S^7$8 on $S^7$9 commuting with $\mathrm{Sp}(2)$0, then one obtains

$\mathrm{Sp}(2)$1

The Gromoll–Meyer example is the motivating model with

$\mathrm{Sp}(2)$2

and it is treated as the paradigmatic instance of a construction in which local orbit geometry is equivariantly related while the global smooth structures differ (Cavenaghi et al., 2024).

This viewpoint was abstracted further in the pullback and cross-diagram framework. The original Gromoll–Meyer construction is encoded by

$\mathrm{Sp}(2)$3

together with the second free action

$\mathrm{Sp}(2)$4

whose quotient is the exotic sphere $\mathrm{Sp}(2)$5. In local trivializations, $\mathrm{Sp}(2)$6 with both actions is equivariantly diffeomorphic to

$\mathrm{Sp}(2)$7

so that the clutching function of $\mathrm{Sp}(2)$8 is

$\mathrm{Sp}(2)$9

This cross-diagram mechanism was pulled back along equivariant maps to construct analogous quotient models for exotic $\mathrm{Sp}(1)\cong S^3$0- and $\mathrm{Sp}(1)\cong S^3$1-spheres, with total spaces $\mathrm{Sp}(1)\cong S^3$2 and $\mathrm{Sp}(1)\cong S^3$3, and quotients diffeomorphic to the only exotic sphere of dimension $\mathrm{Sp}(1)\cong S^3$4 and a generator of the index $\mathrm{Sp}(1)\cong S^3$5 subgroup of $\mathrm{Sp}(1)\cong S^3$6-dimensional homotopy spheres (Sperança, 2010).

The same pullback perspective also clarifies what is structural in the original $\mathrm{Sp}(1)\cong S^3$7-dimensional example. The Gromoll–Meyer sphere is not treated as an isolated biquotient accident, but as the prototype of a wider bundle-plus-commuting-action mechanism in which exoticity is encoded equivariantly (Sperança, 2010).

3. Riemannian metrics, curvature, and extrinsic realizations

For the bi-invariant metric on $\mathrm{Sp}(1)\cong S^3$8, the quotient projection to $\mathrm{Sp}(1)\cong S^3$9 is a Riemannian submersion, so $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$0 carries a metric of nonnegative sectional curvature. This is the classical Gromoll–Meyer result, and the sphere is stressed as the first exotic sphere with nonnegative sectional curvature (Qian et al., 2019). Later, Petersen and Wilhelm proved that there is a metric on the Gromoll–Meyer sphere with positive sectional curvature (0805.0812).

A complementary intrinsic analysis studies a $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$1-parameter family of left-invariant metrics $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$2 on $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$3, defined by

$$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$4

For this family,

$$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$5

and

$$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$6

In an extrinsic extension of the quotient picture, copies of $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$7 appear as isoparametric hypersurfaces $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$8, and their quotients $$\mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\},$$9 are all diffeomorphic to the Gromoll–Meyer sphere. For each $a,b,c,d\in\mathbb H$0, the induced metric on $a,b,c,d\in\mathbb H$1 has positive Ricci curvature and quasi-positive sectional curvature simultaneously (Qian et al., 2019).

A separate Kaluza–Klein Ansatz realizes the Gromoll–Meyer sphere as the Milnor $a,b,c,d\in\mathbb H$2 $a,b,c,d\in\mathbb H$3-bundle over $a,b,c,d\in\mathbb H$4, with a round $a,b,c,d\in\mathbb H$5 as base, unit $a,b,c,d\in\mathbb H$6 as fibre, and $a,b,c,d\in\mathbb H$7 and $a,b,c,d\in\mathbb H$8 $a,b,c,d\in\mathbb H$9 instantons as gauge fields. At a distinguished point

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$0

in the equal-size $\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$1 instanton moduli, combined with the maximally symmetric $\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$2 instanton, the resulting metric has maximal isometry

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$3

For this family the Ricci tensor is computed explicitly, and the detailed derivation gives the condition

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$4

to ensure positive Ricci curvature (Berman et al., 2024).

4. Transnormal, isoparametric, and sub-Riemannian structures

The Gromoll–Meyer sphere also provides a sharp distinction between transnormal and isoparametric behavior. On $\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$5, the function

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$6

is a properly isoparametric function for a certain left-invariant metric, satisfying

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$7

Its focal submanifolds are

$\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$8

Because $\pi:\mathrm{Sp}(2)\to \mathrm{S}^7$9 is invariant under the $\mathrm{SU}(2)$0-action

$\mathrm{SU}(2)$1

it descends to a function $\mathrm{SU}(2)$2 on $\mathrm{SU}(2)$3 with

$\mathrm{SU}(2)$4

However, the projected function is properly transnormal but not isoparametric, with two points as the focal varieties, and the regular level hypersurfaces of $\mathrm{SU}(2)$5 have non-constant mean curvature (Ge et al., 2010).

A different geometric structure arises from principal-bundle horizontality. Using the realization

$\mathrm{SU}(2)$6

with $\mathrm{SU}(2)$7 diagonally embedded in $\mathrm{SU}(2)$8, the horizontal bundle of the larger $\mathrm{SU}(2)$9-bundle over $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$0 descends to a rank-$\mathrm{SU}(2)\cong \mathrm{Sp}(1)$1, co-dimension $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$2 distribution $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$3. The main theorem is that this sub-bundle is completely non-holonomic and of step $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$4 (Bauer et al., 2016).

In the nested principal-bundle formulation, the same geometry is expressed by

$\mathrm{SU}(2)\cong \mathrm{Sp}(1)$5

where $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$6. The induced distribution

$\mathrm{SU}(2)\cong \mathrm{Sp}(1)$7

on $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$8 is bracket generating of step $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$9, and the submersion $7$00 is an $7$01-bundle over $7$02 which is not a principal bundle (Molina et al., 2020).

5. Lorentzian and spacetime geometry

The commuting-action picture has recently been pushed into semi-Riemannian geometry. Starting from the $7$03-diagram with total space $7$04, the standard sphere $7$05 and the exotic sphere $7$06 are equipped with induced $7$07-actions. On $7$08, the action is

$7$09

for $7$10; on the exotic side, using the parametrization

$7$11

the induced action is

$7$12

A sufficiently negative $7$13-Cheeger deformation in the orbit direction yields Lorentzian metrics on the regular strata, and the paper calls the resulting structure “almost Lorentzian” when fixed points force degeneration before repair (Cavenaghi et al., 2024).

The general comparison theorem says that for a $7$14-bundle there is a $7$15-invariant metric on $7$16 such that $7$17 and $7$18 are isometric as metric spaces. Applied to the Gromoll–Meyer diagram, this yields Lorentzian structures on $7$19 and $7$20 with the same quotient geometry under the corresponding circle actions. The main corollary states that the classical sphere $7$21 and the Gromoll–Meyer exotic sphere $7$22 admit time-oriented Lorentzian metrics of positive Ricci curvature with isometric semi-free actions of $7$23 on an open and dense subset, that these actions fix some points out of the regular stratum, and that the orbit-spaces $7$24 and $7$25 are isometric as metric spaces. The same framework also yields Lorentzian metrics with complete space and time-like geodesics on both manifolds (Cavenaghi et al., 2024).

6. Extensions, analytic models, and broader significance

The Gromoll–Meyer sphere functions as a prototype beyond its own dimension. Pulling back the original $7$26 principal bundle along suitable equivariant maps produces quotient models for exotic $7$27- and $7$28-spheres, and in an orthogonal variant also for Kervaire manifolds and Kervaire spheres. This identifies the $7$29-dimensional construction as the model case of a more general quotient-of-bundle mechanism rather than a one-off biquotient phenomenon (Sperança, 2010).

It has also supported analytic and probabilistic constructions. In one approach, isometric stochastic flows of a Stratonovich stochastic differential equation are first constructed on the standard sphere $7$30, realized as $7$31 under the $7$32-action, and then transported to the Gromoll–Meyer sphere $7$33, realized as $7$34 under the $7$35-action, by a homeomorphism $7$36. The induced flow, the Fokker–Planck equation, and the entropy functional can then be related closely across the two topological $7$37-spheres, even though their differential structures are inequivalent (Nurfarisha et al., 2019).

A related but more indirect connection appears in the theory of special generic maps. Saeki’s theorem states that for $7$38 and $7$39, a homotopy $7$40-sphere admits a standard special generic map into $7$41 if and only if it lies in the corresponding level of the Gromoll filtration. That paper does not explicitly mention the Gromoll–Meyer sphere and does not compute its class in $7$42, but it gives a sharp criterion that would decide, once that class is located in the filtration, for which $7$43 the Gromoll–Meyer sphere admits standard special generic maps (Saeki, 2023).

Taken together, these developments present the Gromoll–Meyer sphere as more than a distinguished exotic $7$44-sphere. It is simultaneously a quotient of $7$45, a paradigmatic $7$46-diagram, a test case for positive curvature, an ambient transnormal hypersurface, a co-dimension $7$47 step-$7$48 sub-Riemannian manifold, and a model object in Lorentzian and Kaluza–Klein geometry. Its enduring role comes from the fact that the same commuting-action construction supports all of these viewpoints while preserving the central topological fact: the manifold is a sphere topologically, but not smoothly.