Homogeneous and inhomogeneous isoparametrichypersurfaces in symmetric spaces of noncompact type

Miguel Domınguez Vazquez

Universidade de Santiago de Compostela − Spain

Workshop on the Isoparametric Theory 2019Beijing Normal University

Main new results

Joint work with J. Carlos Dıaz-Ramos and Alberto Rodrıguez-Vazquez

Classification of cohomogeneity one actions on HHn

Uncountably many inhomogeneous isoparametric families ofhypersurfaces with constant principal curvatures

Main new results

Joint work with J. Carlos Dıaz-Ramos and Alberto Rodrıguez-Vazquez

Classification of cohomogeneity one actions on HHn

Uncountably many inhomogeneous isoparametric families ofhypersurfaces with constant principal curvatures

Main new results

Joint work with J. Carlos Dıaz-Ramos and Alberto Rodrıguez-Vazquez

Classification of cohomogeneity one actions on HHn

=⇒ Classification of cohomogeneity one actions onsymmetric spaces of rank one

Uncountably many inhomogeneous isoparametric families ofhypersurfaces with constant principal curvatures

Main new results

Joint work with J. Carlos Dıaz-Ramos and Alberto Rodrıguez-Vazquez

Classification of cohomogeneity one actions on HHn

=⇒ Classification of cohomogeneity one actions onsymmetric spaces of rank one

Uncountably many inhomogeneous isoparametric families ofhypersurfaces with constant principal curvatures

Contents

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Contents

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Cohomogeneity one actions

M complete Riemannian manifold

Definition

A cohomogeneity one action on M is a proper isometric action on Mwith codimension one orbits.

Cohomogeneity one actions

M complete Riemannian manifold

Definition

A cohomogeneity one action on M is a proper isometric action on Mwith codimension one orbits.

Properties

All the orbits, except at most two, are hypersurfaces.

The orbit space is isometric to S1, [a, b], R or [0,+∞).

Cohomogeneity one actions

M complete Riemannian manifold

Definition

A cohomogeneity one action on M is a proper isometric action on Mwith codimension one orbits.

Properties

All the orbits, except at most two, are hypersurfaces.

The orbit space is isometric to S1, [a, b], R or [0,+∞).

SO(2) � R2

A · v = AvR � R2

t · v = v + tw

SO(2)× R � R3

(A, t) · v =(A 00 1

)v +

(0t

) SO(2) � S2

A · v =(A 00 1

)v

Homogeneous hypersurfaces

M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on M are orbit equivalent ifthere exists ϕ ∈ Isom(M) that maps each G1-orbit to a G2-orbit.

Problem

Classify cohomogeneity one actions on M up to orbit equivalence.

Definition

A submanifold is a homogeneous submanifold if it is an orbit of anisometric action. Homogeneous hypersurfaces are precisely thecodimension one orbits of cohomogeneity one actions.

Equivalent problem

Classify homogeneous hypersurfaces in a given Riemannian manifold M.

Homogeneous hypersurfaces

M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on M are orbit equivalent ifthere exists ϕ ∈ Isom(M) that maps each G1-orbit to a G2-orbit.

Problem

Classify cohomogeneity one actions on M up to orbit equivalence.

Definition

A submanifold is a homogeneous submanifold if it is an orbit of anisometric action. Homogeneous hypersurfaces are precisely thecodimension one orbits of cohomogeneity one actions.

Equivalent problem

Classify homogeneous hypersurfaces in a given Riemannian manifold M.

Homogeneous hypersurfaces

M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on M are orbit equivalent ifthere exists ϕ ∈ Isom(M) that maps each G1-orbit to a G2-orbit.

Problem

Classify cohomogeneity one actions on M up to orbit equivalence.

Definition

A submanifold is a homogeneous submanifold if it is an orbit of anisometric action. Homogeneous hypersurfaces are precisely thecodimension one orbits of cohomogeneity one actions.

Equivalent problem

Classify homogeneous hypersurfaces in a given Riemannian manifold M.

Homogeneous hypersurfaces

M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on M are orbit equivalent ifthere exists ϕ ∈ Isom(M) that maps each G1-orbit to a G2-orbit.

Problem

Classify cohomogeneity one actions on M up to orbit equivalence.

Definition

A submanifold is a homogeneous submanifold if it is an orbit of anisometric action. Homogeneous hypersurfaces are precisely thecodimension one orbits of cohomogeneity one actions.

Equivalent problem

Classify homogeneous hypersurfaces in a given Riemannian manifold M.

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Classification of cohomogeneity one actions

Cohomogeneity one actions have been classified, up to orbit equivalence, in

Euclidean spaces Rn [Somigliana (1918), Segre (1938)]

Real hyperbolic spaces RHn [Cartan (1939)]

Round spheres Sn [Hsiang, Lawson (1971), Takagi, Takahashi (1972)]

Complex projective spaces CPn [Takagi (1973)]

Quaternionic projective spaces HPn [D’Atri (1979), Iwata (1978)]

Cayley projective plane OP2 [Iwata (1981)]

Irreducible symmetric spaces of compact type [Kollross (2002)]

S2 × S2 [Urbano (2016)]

Homogeneous 3-manifolds with 4-dimensional isometry group(E(κ, τ)-spaces) [DV, Manzano (2018)]

Question

What happens in symmetric spaces of noncompact type?

Isoparametric hypersurfaces

M Riemannian manifold

Definition [Levi-Civita (1937)]

A hypersurface M in M is isoparametric if M and its nearby equidistanthypersurfaces have constant mean curvature.

Equivalently, if M is a regular level set of a function f : Uopen

⊂ M → R suchthat |∇f | = a ◦ f and ∆f = b ◦ f , for smooth functions a, b.

Isoparametric hypersurfaces

M Riemannian manifold

Definition [Levi-Civita (1937)]

A hypersurface M in M is isoparametric if M and its nearby equidistanthypersurfaces have constant mean curvature.

Equivalently, if M is a regular level set of a function f : Uopen

⊂ M → R suchthat |∇f | = a ◦ f and ∆f = b ◦ f , for smooth functions a, b.

Isoparametric hypersurfaces

M Riemannian manifold

Definition [Levi-Civita (1937)]

A hypersurface M in M is isoparametric if M and its nearby equidistanthypersurfaces have constant mean curvature.

Equivalently, if M is a regular level set of a function f : Uopen

⊂ M → R suchthat |∇f | = a ◦ f and ∆f = b ◦ f , for smooth functions a, b.

Isoparametric hypersurfaces

M Riemannian manifold

Definition [Levi-Civita (1937)]

A hypersurface M in M is isoparametric if M and its nearby equidistanthypersurfaces have constant mean curvature.

Equivalently, if M is a regular level set of a function f : Uopen

⊂ M → R suchthat |∇f | = a ◦ f and ∆f = b ◦ f , for smooth functions a, b.

M homogeneoushypersurface

⇓M isoparametric

hypersurfacewith constant

principal curvatures

Isoparametric hypersurfaces in space forms

Theorem [Cartan (1939), Segre (1938)]

Let M be a hypersurface in a real space form M ∈ {Rn,RHn, Sn}. Then:

M is isoparametric ⇔ M has constant principal curvatures

If M ∈ {Rn,RHn}, M is isoparametric ⇔ M is homogeneous

Isoparametric hypersurfaces in space forms

Theorem [Cartan (1939), Segre (1938)]

Let M be a hypersurface in a real space form M ∈ {Rn,RHn, Sn}. Then:

M is isoparametric ⇔ M has constant principal curvatures

If M ∈ {Rn,RHn}, M is isoparametric ⇔ M is homogeneous

Classification in the Euclidean space Rn [Segre (1938)]

Parallel hyperplanesRn−1

Concentric spheresSn−1

Generalized cylindersSk × Rn−k−1

Isoparametric hypersurfaces in space forms

Theorem [Cartan (1939), Segre (1938)]

Let M be a hypersurface in a real space form M ∈ {Rn,RHn, Sn}. Then:

M is isoparametric ⇔ M has constant principal curvatures

If M ∈ {Rn,RHn}, M is isoparametric ⇔ M is homogeneous

Classification in the real hyperbolic space RHn [Cartan (1939)]

Tot. geod. RHn−1

and equidistanthypersurfaces

Tubes around atot. geod. RHk

Geodesic spheres Horospheres

Isoparametric hypersurfaces in space forms

Theorem [Cartan (1939), Segre (1938)]

Let M be a hypersurface in a real space form M ∈ {Rn,RHn, Sn}. Then:

M is isoparametric ⇔ M has constant principal curvatures

If M ∈ {Rn,RHn}, M is isoparametric ⇔ M is homogeneous

Classification in spheres Sn

There are inhomogeneous examples [Ferus, Karcher, Munzner(1981)]

All isoparametric hypersurfaces are homogeneous or ofFKM-type [Cartan; Munzner; Takagi; Ozeki, Takeuchi; Tang; Fang;Stolz; Cecil, Chi, Jensen; Immervoll; Abresch; Dorfmeister, Neher;Miyaoka; Chi]

Isoparametric hypersurfaces in nonconstant curvature

General geometric and topological structure results [Wang (1987),Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge,Radeschi (2015), Ge (2016)]

Classification in CPn, n 6= 15 [DV (2016)] and in HPn, n 6= 7 [DV,Gorodski (2018)]

There are countably many inhomogeneous examples, all of them withnonconstant principal curvatures

Classification in S2 × S2 [Urbano (2016)] and in E(κ, τ)-spaces [DV,Manzano (2018)]

In these two cases, all examples are homogeneous

Isoparametric hypersurfaces in nonconstant curvature

General geometric and topological structure results [Wang (1987),Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge,Radeschi (2015), Ge (2016)]

Classification in CPn, n 6= 15 [DV (2016)] and in HPn, n 6= 7 [DV,Gorodski (2018)]

There are countably many inhomogeneous examples, all of them withnonconstant principal curvatures

Classification in S2 × S2 [Urbano (2016)] and in E(κ, τ)-spaces [DV,Manzano (2018)]

In these two cases, all examples are homogeneous

Isoparametric hypersurfaces in nonconstant curvature

General geometric and topological structure results [Wang (1987),Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge,Radeschi (2015), Ge (2016)]

Classification in CPn, n 6= 15 [DV (2016)] and in HPn, n 6= 7 [DV,Gorodski (2018)]

There are countably many inhomogeneous examples, all of them withnonconstant principal curvatures

Classification in S2 × S2 [Urbano (2016)] and in E(κ, τ)-spaces [DV,Manzano (2018)]

In these two cases, all examples are homogeneous

Isoparametric hypersurfaces in nonconstant curvature

General geometric and topological structure results [Wang (1987),Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge,Radeschi (2015), Ge (2016)]

Classification in CPn, n 6= 15 [DV (2016)] and in HPn, n 6= 7 [DV,Gorodski (2018)]

There are countably many inhomogeneous examples, all of them withnonconstant principal curvatures

Classification in S2 × S2 [Urbano (2016)] and in E(κ, τ)-spaces [DV,Manzano (2018)]

In these two cases, all examples are homogeneous

Question

What happens in symmetric spaces of noncompact type?

Symmetric spaces of noncompact type

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces are complete and homogeneous

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces are complete and homogeneous

M ∼= G/K , where G = Isom0(M) and K = {g ∈ G : g(o) = o} are Liegroups, and o ∈ M is a base point

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces are complete and homogeneous

M ∼= G/K , where G = Isom0(M) and K = {g ∈ G : g(o) = o} are Liegroups, and o ∈ M is a base point

compact type noncompact type Euclidean typeduality

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold M whose geodesicsymmetry σp : expp(v) 7→ expp(−v), v ∈ TpM, around each p ∈ M is aglobal isometry of M.

Symmetric spaces are complete and homogeneous

M ∼= G/K , where G = Isom0(M) and K = {g ∈ G : g(o) = o} are Liegroups, and o ∈ M is a base point

compact type noncompact type Euclidean typeduality

M compact,sec(M) ≥ 0,

g compact semisimple

M noncompact,sec(M) ≤ 0,g noncompact

semisimple

M = Rn/Γ flat

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type

=⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type

M ∼= G/K symmetric space of noncompact type =⇒ Mdiffeo.∼= Bn

g = k⊕ p Cartan decomposition, p ∼= ToM

a maximal abelian subspace of p, rank M := dim a

Iwasawa decomposition

g = k⊕ a⊕ nn is nilpotent

Gdiffeo.∼= K × A× N K A N

a⊕ n Lie subalgebra of g ; AN Lie subgroup of G

AN acts freely and transitively on M ; AN is diffeomorphic to M

The solvable model of a symmetric space of noncompact type

M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type and rank one

M ∼= G/Kisom.∼= AN symmetric space of noncompact type, rank M = 1

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

Symmetric spaces of noncompact type and rank 1

Symmetric spaces of noncompact type and rank one

M ∼= G/Kisom.∼= AN symmetric space of noncompact type, rank M = 1

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

Symmetric spaces of noncompact type and rank 1

Symmetric spaces of noncompact type and rank one

M ∼= G/Kisom.∼= AN symmetric space of noncompact type, rank M = 1

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

Symmetric spaces of noncompact type and rank 1

MRHn CHn HHn OH2

SO0(1,n)SO(n)

SU(1,n)S(U(1)×U(n))

Sp(1,n)Sp(1)×Sp(n)

F−204

Spin(9)

v Rn−1 Cn−1 Hn−1 Odim z 0 1 3 7

Symmetric spaces of noncompact type

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Cohomogeneity one actions on hyperbolic spaces

FHn symmetric space of noncompact type and rank one, F ∈ {R,C,H,O}

Cohomogeneity one actions with a totally geodesic singular orbit[Berndt, Bruck (2001)]

Tubes around totally geodesic submanifolds P in FHn are homogeneous ifand only if

in RHn: P = {point},RH1, . . . ,RHn−1

in CHn: P = {point},CH1, . . . ,CHn−1,RHn

in HHn: P = {point},HH1, . . . ,HHn−1,CHn

in OH2: P = {point},OH1,HH2

P

Cohomogeneity one actions on hyperbolic spaces

FHn ∼= G/Kisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Symmetric spaces of noncompact type and rank 1

FHn RHn CHn HHn OH2

v Rn−1 Cn−1 Hn−1 O

Cohomogeneity one actions on hyperbolic spaces

FHn ∼= G/Kisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Symmetric spaces of noncompact type and rank 1

FHn RHn CHn HHn OH2

v Rn−1 Cn−1 Hn−1 O

Cohomogeneity one actions without singular orbits [Berndt, Bruck(2001), Berndt, Tamaru (2003)]

Orbit equivalent to the action of:

N ; horosphere foliation

The connected subgroup of Gwith Lie algebra a⊕w⊕ z, wherew is a (real) hyperplane in v

Cohomogeneity one actions on hyperbolic spaces

FHn ∼= G/Kisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Symmetric spaces of noncompact type and rank 1

FHn RHn CHn HHn OH2

v Rn−1 Cn−1 Hn−1 O

Cohomogeneity one actions with a non-totally singular orbit [Berndt,Bruck (2001)]

w ( v (real) subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

The tubes around Sw are homogeneous if and only ifNK0(w) acts transitively on the unit sphere of w⊥

(the orthogonal complement of w in v)

Cohomogeneity one actions on hyperbolic spaces

FHn ∼= G/Kisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Symmetric spaces of noncompact type and rank 1

FHn RHn CHn HHn OH2

v Rn−1 Cn−1 Hn−1 O

Cohomogeneity one actions with a non-totally singular orbit [Berndt,Bruck (2001)]

w ( v (real) subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

The tubes around Sw are homogeneous if and only ifNK0(w) acts transitively on the unit sphere of w⊥

(the orthogonal complement of w in v)

Sw

tube

Cohomogeneity one actions on hyperbolic spaces

FHn ∼= G/Kisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Symmetric spaces of noncompact type and rank 1

FHn RHn CHn HHn OH2

v Rn−1 Cn−1 Hn−1 O

Cohomogeneity one actions with a non-totally singular orbit [Berndt,Bruck (2001)]

w ( v (real) subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

The tubes around Sw are homogeneous if and only ifNK0(w) acts transitively on the unit sphere of w⊥

(the orthogonal complement of w in v)

Sw

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit.

Its orbit foliation is regular.

There is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit.

Its orbit foliation is regular.

There is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. X

Its orbit foliation is regular.

There is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. XIts orbit foliation is regular.

There is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. XIts orbit foliation is regular. X

There is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. XIts orbit foliation is regular. XThere is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. XIts orbit foliation is regular. XThere is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

The study of the last case was carried out for RHn, CHn, HH2 and OH2

Cohomogeneity one actions on hyperbolic spaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n), K0 = NK (a)

Theorem [Berndt, Tamaru (2007)]

For a cohomogeneity one action on FHn, one of the following holds:

There is a totally geodesic singular orbit. XIts orbit foliation is regular. XThere is a non-totally geodesic singular orbit Sw, where w ( v is suchthat NK0(w) acts transitively on the unit sphere of w⊥.

The study of the last case was carried out for RHn, CHn, HH2 and OH2

Problem

Analyze the last case for HHn, n ≥ 3, to conclude the classification.

Symmetric spaces of noncompact type

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

Sw

tube

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

Sw

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

Sw

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

New isoparametric hypersurfaces

FHnisom.∼= AN symmetric space of noncompact type and rank one

a⊕ n = a⊕ v⊕ z, a ∼= R, z = Z (n)

New isoparametric hypersurfaces [Dıaz-Ramos, DV (2013)]

w ( v real subspace =⇒ sw = a⊕w⊕ z is a Liealgebra

Sw connected subgroup of AN with Lie algebra sw

Sw is a homogeneous minimal submanifold

The tubes around Sw are isoparametric

Sw

In RHn such hypersurfaces are homogeneous

In CHn and HHn, n ≥ 3, there are inhomogeneous isoparametricfamilies of hypersurfaces with nonconstant principal curvatures

In OH2 there is one inhomogeneous isoparametric family ofhypersurfaces with constant principal curvatures (when dimw = 3)

Classification in the complex hyperbolic space

Theorem [Dıaz-Ramos, DV, Sanmartın-Lopez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn isisoparametric if and only if it is an open subset of:

A tube around a totally geodesic complex hyperbolic space CHk

A tube around a totally geodesic real hyperbolic space RHn

A horosphere

A tube around a homogeneous minimal submanifold Sw

Classification in the complex hyperbolic space

Theorem [Dıaz-Ramos, DV, Sanmartın-Lopez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn isisoparametric if and only if it is an open subset of:

A tube around a totally geodesic complex hyperbolic space CHk

A tube around a totally geodesic real hyperbolic space RHn

A horosphere

A tube around a homogeneous minimal submanifold Sw

Classical examples [Montiel (1985)]: all are homogeneous

Classification in the complex hyperbolic space

Theorem [Dıaz-Ramos, DV, Sanmartın-Lopez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn isisoparametric if and only if it is an open subset of:

A tube around a totally geodesic complex hyperbolic space CHk

A tube around a totally geodesic real hyperbolic space RHn

A horosphere

A tube around a homogeneous minimal submanifold Sw

Classical examples [Montiel (1985)]: all are homogeneous

New examples: there are both (uncountably many) homogeneous[Berndt, Bruck (2001)] and inhomogeneous [Dıaz-Ramos, DV (2012)]examples, depending on w ⊂ v

The quaternionic hyperbolic space

1 Homogeneous and isoparametric hypersurfaces2 Symmetric spaces of noncompact type and rank one

1 Cohomogeneity one actions2 Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

The quaternionic hyperbolic space

Problem

Classify cohomogeneity one actions on HHn+1, n ≥ 2.

Equivalent problem [Berndt, Tamaru (2007)]

Classify real subspaces w ⊂ v ∼= Hn such that NK0(w) acts transitively onthe unit sphere of w⊥.

K0∼= Sp(n)Sp(1) acts on v ∼= Hn via (A, q) · v = Avq−1

Definition

A real subspace V of Hn is protohomogeneous if there is a (connected)subgroup of Sp(n)Sp(1) that acts transitively on the unit sphere of V .

Equivalent problem

Classify protohomogeneous subspaces of Hn.

The quaternionic hyperbolic space

Problem

Classify cohomogeneity one actions on HHn+1, n ≥ 2.

Equivalent problem [Berndt, Tamaru (2007)]

Classify real subspaces w ⊂ v ∼= Hn such that NK0(w) acts transitively onthe unit sphere of w⊥.

K0∼= Sp(n)Sp(1) acts on v ∼= Hn via (A, q) · v = Avq−1

Definition

A real subspace V of Hn is protohomogeneous if there is a (connected)subgroup of Sp(n)Sp(1) that acts transitively on the unit sphere of V .

Equivalent problem

Classify protohomogeneous subspaces of Hn.

The quaternionic hyperbolic space

Problem

Classify cohomogeneity one actions on HHn+1, n ≥ 2.

Equivalent problem [Berndt, Tamaru (2007)]

Classify real subspaces w ⊂ v ∼= Hn such that NK0(w) acts transitively onthe unit sphere of w⊥.

K0∼= Sp(n)Sp(1) acts on v ∼= Hn via (A, q) · v = Avq−1

Definition

A real subspace V of Hn is protohomogeneous if there is a (connected)subgroup of Sp(n)Sp(1) that acts transitively on the unit sphere of V .

Equivalent problem

Classify protohomogeneous subspaces of Hn.

The quaternionic hyperbolic space

Problem

Classify cohomogeneity one actions on HHn+1, n ≥ 2.

Equivalent problem [Berndt, Tamaru (2007)]

Classify real subspaces w ⊂ v ∼= Hn such that NK0(w) acts transitively onthe unit sphere of w⊥.

K0∼= Sp(n)Sp(1) acts on v ∼= Hn via (A, q) · v = Avq−1

Definition

A real subspace V of Hn is protohomogeneous if there is a (connected)subgroup of Sp(n)Sp(1) that acts transitively on the unit sphere of V .

Equivalent problem

Classify protohomogeneous subspaces of Hn.

The quaternionic hyperbolic space

Problem

Classify cohomogeneity one actions on HHn+1, n ≥ 2.

Equivalent problem [Berndt, Tamaru (2007)]

Classify real subspaces w ⊂ v ∼= Hn such that NK0(w) acts transitively onthe unit sphere of w⊥.

K0∼= Sp(n)Sp(1) acts on v ∼= Hn via (A, q) · v = Avq−1

Definition

A real subspace V of Hn is protohomogeneous if there is a (connected)subgroup of Sp(n)Sp(1) that acts transitively on the unit sphere of V .

Equivalent problem

Classify protohomogeneous subspaces of Hn.

Getting intuition in the complex setting

Analogous definition in Cn

A real subspace V of Cn is protohomogeneous if there is a (connected)subgroup of U(n) that acts transitively on the unit sphere of V

Analogous problem in Cn

Classify protohomogeneous subspaces of Cn

{e1, . . . , en} C-orthonormal basis of Cn, J complex structure of Cn

Totally real subspaces V = spanR{e1, . . . , ek} are protohomogeneous; SO(k) ⊂ U(n) acts transitively on Sk−1

Complex subspaces V = spanC{e1, . . . , ek} are protohomogeneous; U(k) ⊂ U(n) acts transitively on S2k−1

V = spanR{e1, Je1, e2} is not protohomogeneous; NU(n)(V ) = U(1)× U(n − 2) does not act transitively on S2

Getting intuition in the complex setting

Analogous definition in Cn

A real subspace V of Cn is protohomogeneous if there is a (connected)subgroup of U(n) that acts transitively on the unit sphere of V

Analogous problem in Cn

Classify protohomogeneous subspaces of Cn

{e1, . . . , en} C-orthonormal basis of Cn, J complex structure of Cn

Totally real subspaces V = spanR{e1, . . . , ek} are protohomogeneous; SO(k) ⊂ U(n) acts transitively on Sk−1

Complex subspaces V = spanC{e1, . . . , ek} are protohomogeneous; U(k) ⊂ U(n) acts transitively on S2k−1

V = spanR{e1, Je1, e2} is not protohomogeneous; NU(n)(V ) = U(1)× U(n − 2) does not act transitively on S2

Getting intuition in the complex setting

Analogous definition in Cn

A real subspace V of Cn is protohomogeneous if there is a (connected)subgroup of U(n) that acts transitively on the unit sphere of V

Analogous problem in Cn

Classify protohomogeneous subspaces of Cn

{e1, . . . , en} C-orthonormal basis of Cn, J complex structure of Cn

Totally real subspaces V = spanR{e1, . . . , ek} are protohomogeneous; SO(k) ⊂ U(n) acts transitively on Sk−1

Complex subspaces V = spanC{e1, . . . , ek} are protohomogeneous; U(k) ⊂ U(n) acts transitively on S2k−1

V = spanR{e1, Je1, e2} is not protohomogeneous; NU(n)(V ) = U(1)× U(n − 2) does not act transitively on S2

Getting intuition in the complex setting

Analogous definition in Cn

A real subspace V of Cn is protohomogeneous if there is a (connected)subgroup of U(n) that acts transitively on the unit sphere of V

Analogous problem in Cn

Classify protohomogeneous subspaces of Cn

{e1, . . . , en} C-orthonormal basis of Cn, J complex structure of Cn

Totally real subspaces V = spanR{e1, . . . , ek} are protohomogeneous; SO(k) ⊂ U(n) acts transitively on Sk−1

Complex subspaces V = spanC{e1, . . . , ek} are protohomogeneous; U(k) ⊂ U(n) acts transitively on S2k−1

V = spanR{e1, Je1, e2} is not protohomogeneous; NU(n)(V ) = U(1)× U(n − 2) does not act transitively on S2

Getting intuition in the complex setting

Analogous definition in Cn

A real subspace V of Cn is protohomogeneous if there is a (connected)subgroup of U(n) that acts transitively on the unit sphere of V

Analogous problem in Cn

Classify protohomogeneous subspaces of Cn

{e1, . . . , en} C-orthonormal basis of Cn, J complex structure of Cn

Totally real subspaces V = spanR{e1, . . . , ek} are protohomogeneous; SO(k) ⊂ U(n) acts transitively on Sk−1

Complex subspaces V = spanC{e1, . . . , ek} are protohomogeneous; U(k) ⊂ U(n) acts transitively on S2k−1

V = spanR{e1, Je1, e2} is not protohomogeneous; NU(n)(V ) = U(1)× U(n − 2) does not act transitively on S2

Getting intuition in the complex setting

V real subspace of Cn, π : Cn → V orthogonal projection, v ∈ V \ {0}

Definition

The Kahler angle of v with respect to V is the angle ϕ ∈ [0, π/2]between Jv and V . Equivalently, 〈πJv , πJv〉 = cos2 ϕ 〈v , v〉.V has constant Kahler angle ϕ if the Kahler angle of any v ∈ V \ {0}with respect to V is ϕ.

Totally real subspaces have constant Kahler angle π/2

Complex subspaces have constant Kahler angle 0

V = spanR{e1, Je1, e2} does not have constant Kahler angle

Proposition [Berndt, Bruck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant Kahler angle.Moreover, V has constant Kahler angle ϕ ∈ [0, π/2) if and only ifV = span{e1, cosϕJe1 + sinϕJe2, . . . , ek , cosϕJe2k−1 + sinϕJe2k}.

Getting intuition in the complex setting

V real subspace of Cn, π : Cn → V orthogonal projection, v ∈ V \ {0}

Definition

The Kahler angle of v with respect to V is the angle ϕ ∈ [0, π/2]between Jv and V . Equivalently, 〈πJv , πJv〉 = cos2 ϕ 〈v , v〉.V has constant Kahler angle ϕ if the Kahler angle of any v ∈ V \ {0}with respect to V is ϕ.

Totally real subspaces have constant Kahler angle π/2

Complex subspaces have constant Kahler angle 0

V = spanR{e1, Je1, e2} does not have constant Kahler angle

Proposition [Berndt, Bruck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant Kahler angle.Moreover, V has constant Kahler angle ϕ ∈ [0, π/2) if and only ifV = span{e1, cosϕJe1 + sinϕJe2, . . . , ek , cosϕJe2k−1 + sinϕJe2k}.

Getting intuition in the complex setting

V real subspace of Cn, π : Cn → V orthogonal projection, v ∈ V \ {0}

Definition

The Kahler angle of v with respect to V is the angle ϕ ∈ [0, π/2]between Jv and V . Equivalently, 〈πJv , πJv〉 = cos2 ϕ 〈v , v〉.V has constant Kahler angle ϕ if the Kahler angle of any v ∈ V \ {0}with respect to V is ϕ.

Totally real subspaces have constant Kahler angle π/2

Complex subspaces have constant Kahler angle 0

V = spanR{e1, Je1, e2} does not have constant Kahler angle

Proposition [Berndt, Bruck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant Kahler angle.

Moreover, V has constant Kahler angle ϕ ∈ [0, π/2) if and only ifV = span{e1, cosϕJe1 + sinϕJe2, . . . , ek , cosϕJe2k−1 + sinϕJe2k}.

Getting intuition in the complex setting

V real subspace of Cn, π : Cn → V orthogonal projection, v ∈ V \ {0}

Definition

The Kahler angle of v with respect to V is the angle ϕ ∈ [0, π/2]between Jv and V . Equivalently, 〈πJv , πJv〉 = cos2 ϕ 〈v , v〉.V has constant Kahler angle ϕ if the Kahler angle of any v ∈ V \ {0}with respect to V is ϕ.

Totally real subspaces have constant Kahler angle π/2

Complex subspaces have constant Kahler angle 0

V = spanR{e1, Je1, e2} does not have constant Kahler angle

Proposition [Berndt, Bruck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant Kahler angle.Moreover, V has constant Kahler angle ϕ ∈ [0, π/2) if and only ifV = span{e1, cosϕJe1 + sinϕJe2, . . . , ek , cosϕJe2k−1 + sinϕJe2k}.

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

J ⊂ EndR(Hn) quaternionic structure of Hn

{J1, J2, J3} canonical basis of J: J2i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

V real subspace of Hn, v ∈ V \ {0}, π : Hn → V orthogonal projection

Definition

Consider the symmetric bilinear form

Lv : J× J→ R, Lv (J, J ′) := 〈πJv , πJ ′v〉.

The quaternionic Kahler angle of v with respect to V is the triple(ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv arecos2 ϕi 〈v , v〉, i = 1, 2, 3.

There is a canonical basis {J1, J2, J3} of J made of eigenvectors of Lv

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

J ⊂ EndR(Hn) quaternionic structure of Hn

{J1, J2, J3} canonical basis of J: J2i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

V real subspace of Hn, v ∈ V \ {0}, π : Hn → V orthogonal projection

Definition

Consider the symmetric bilinear form

Lv : J× J→ R, Lv (J, J ′) := 〈πJv , πJ ′v〉.

The quaternionic Kahler angle of v with respect to V is the triple(ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv arecos2 ϕi 〈v , v〉, i = 1, 2, 3.

There is a canonical basis {J1, J2, J3} of J made of eigenvectors of Lv

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

J ⊂ EndR(Hn) quaternionic structure of Hn

{J1, J2, J3} canonical basis of J: J2i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

V real subspace of Hn, v ∈ V \ {0}, π : Hn → V orthogonal projection

Definition

Consider the symmetric bilinear form

Lv : J× J→ R, Lv (J, J ′) := 〈πJv , πJ ′v〉.

The quaternionic Kahler angle of v with respect to V is the triple(ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv arecos2 ϕi 〈v , v〉, i = 1, 2, 3.

There is a canonical basis {J1, J2, J3} of J made of eigenvectors of Lv

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

J ⊂ EndR(Hn) quaternionic structure of Hn

{J1, J2, J3} canonical basis of J: J2i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

V real subspace of Hn, v ∈ V \ {0}, π : Hn → V orthogonal projection

Definition

Consider the symmetric bilinear form

Lv : J× J→ R, Lv (J, J ′) := 〈πJv , πJ ′v〉.

The quaternionic Kahler angle of v with respect to V is the triple(ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv arecos2 ϕi 〈v , v〉, i = 1, 2, 3.

There is a canonical basis {J1, J2, J3} of J made of eigenvectors of Lv

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

J ⊂ EndR(Hn) quaternionic structure of Hn

{J1, J2, J3} canonical basis of J: J2i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

V real subspace of Hn, v ∈ V \ {0}, π : Hn → V orthogonal projection

Definition

Consider the symmetric bilinear form

Lv : J× J→ R, Lv (J, J ′) := 〈πJv , πJ ′v〉.

The quaternionic Kahler angle of v with respect to V is the triple(ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv arecos2 ϕi 〈v , v〉, i = 1, 2, 3.

There is a canonical basis {J1, J2, J3} of J made of eigenvectors of Lv

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Back to the quaternionic setting

Proposition [Berndt, Bruck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic Kahler angle.

There are subspaces V with constant quaternionic Kahler angle (0, 0, 0),(0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (ϕ, π/2, π/2), (0, ϕ, ϕ)...

But not every triple arises as the constant quaternionic Kahler angle of asubspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic Kahler angle.

Question

Does constant quaternionic Kahler angle imply protohomogeneous?

Theorem [Dıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only ifw⊥ ⊂ v has constant quaternionic Kahler angle.

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k

⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V ,

π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto V

v ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).

X

Case k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).XCase k ≡ 0 (mod 4).

?

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dimV = k⇒ V constant quaternionic Kahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Sk−1 unit sphere of V , π : Hn → V orthogonal projection onto Vv ∈ Sk−1 ⇒ 〈Jv , v〉 = 0 ⇒ 〈πJv , v〉 = 0 for any J ∈ J

∆v := {πJv : J ∈ J} smooth distribution on Sk−1, rank ∆ ∈ {0, 1, 2, 3}

Applying the generalized hairy ball theorem [Adams (1963)]

If k ≥ 5 is odd, then Φ(V ) = (π/2, π/2, π/2).

If k ≡ 2 (mod 4), then Φ(V ) = (ϕ, π/2, π/2), for some ϕ ∈ [0, π/2].

If k = 3, then Φ(V ) = (ϕ,ϕ, π/2), for some ϕ ∈ [0, π/2].

Remaining cases

Classify subspaces V with k = 3 and Φ(V ) = (ϕ,ϕ, π/2).XCase k ≡ 0 (mod 4). ?

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5.

For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J

(i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji )

such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V .

Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dimV = k = 4r

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

Assume k ≥ 5. For simplicity, assume ϕ3 6= π/2.

1 There exists a canonical basis {J1, J2, J3} of J (i.e. J2i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji ) such that the Kahler angle of any v ∈ Sk−1

with respect to V and the complex structure Ji is ϕi .

2 Define Pi = 1cosϕi

πJi : V → V . Then PiPj + PjPi = −2δij Id.

3 {P1,P2,P3} induces a structure of Cl(3)-module on V .

4 V =(⊕

V+

)⊕(⊕

V−), where V+ and V− are the two

inequivalent irreducible Cl(3)-modules, dimV± = 4.

5 Each factor has constant quaternionic Kahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

Protohomogeneous subspaces in Hn

V protohomogeneous subspace of Hn, dimV = 4r , Φ(V ) = (ϕ1, ϕ2, ϕ3)

V =(⊕

V+

)⊕(⊕

V−)

V+ and V− two inequivalent irreducible Cl(3)-modulesdimV± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

6 There are two types of subspaces V of dimension 4:

V+, which exists if and only if cosϕ1 + cosϕ2 + cosϕ3 ≤ 1.V−, which exists if and only if cosϕ1 + cosϕ2 − cosϕ3 ≤ 1.

7 6 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

8 If V , with dimV = 4r , then either V =⊕

V+ or V =⊕

V−.

From this, one can obtain the classification of protohomogeneoussubspaces of Hn, and hence of cohomogeneity one actions on HHn+1.

Question

What if we mix both types of 4-dimensional subspaces, V+ and V−?

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.

New isoparametric hypersurfaces

V =

( r+⊕V+

)⊕( r−⊕

V−

)V+ and V− two inequivalent irreducible Cl(3)-modules, dimV± = 4Φ(V±) = (ϕ1, ϕ2, ϕ3) with cosϕ1 + cosϕ2 + cosϕ3 ≤ 1

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn withconstant quaternionic Kahler angle.

HHn+1isom.∼= AN, a⊕ n = a⊕ v⊕ z, v ∼= Hn

w := orthogonal complement of V in vsw = a⊕w⊕ z ; Sw connected subgroup of AN

Theorem [Dıaz-Ramos, DV, Rodrıguez-Vazquez (2019)]

Sw and the tubes around it define an inhomogeneous isoparametricfamily of hypersurfaces with constant principal curvatures in HHn+1.