Homogeneous and inhomogeneous isoparametric hypersurfaces in - - PowerPoint PPT Presentation

Homogeneous and inhomogeneous isoparametric hypersurfaces in symmetric spaces of noncompact type

Miguel Dom´ ınguez V´ azquez

Universidade de Santiago de Compostela − Spain Workshop on the Isoparametric Theory 2019 Beijing Normal University

Main new results

Joint work with J. Carlos D´ ıaz-Ramos and Alberto Rodr´ ıguez-V´ azquez

Main new results

Joint work with J. Carlos D´ ıaz-Ramos and Alberto Rodr´ ıguez-V´ azquez Classification of cohomogeneity one actions on HHn

Main new results

Joint work with J. Carlos D´ ıaz-Ramos and Alberto Rodr´ ıguez-V´ azquez Classification of cohomogeneity one actions on HHn = ⇒ Classification of cohomogeneity one actions on symmetric spaces of rank one

Main new results

Joint work with J. Carlos D´ ıaz-Ramos and Alberto Rodr´ ıguez-V´ azquez Classification of cohomogeneity one actions on HHn = ⇒ Classification of cohomogeneity one actions on symmetric spaces of rank one Uncountably many inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures

Contents

1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Contents

1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

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 ¯ M with codimension one orbits.

Cohomogeneity one actions

¯ M complete Riemannian manifold

Definition

A cohomogeneity one action on ¯ M is a proper isometric action on ¯ M with 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 ¯ M with 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 = Av R R2 t · v = v + tw SO(2) × R R3 (A, t) · v = A

1

• v +

t

• SO(2) S2

A · v = A

1

• v

Homogeneous hypersurfaces

¯ M complete Riemannian manifold

Definition

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

Homogeneous hypersurfaces

¯ M complete Riemannian manifold

Definition

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

Problem

Classify cohomogeneity one actions on ¯ M up to orbit equivalence.

Homogeneous hypersurfaces

¯ M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on ¯ M are orbit equivalent if there 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 an isometric action. Homogeneous hypersurfaces are precisely the codimension one orbits of cohomogeneity one actions.

Homogeneous hypersurfaces

¯ M complete Riemannian manifold

Definition

Two isometric actions of groups G1, G2 on ¯ M are orbit equivalent if there 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 an isometric action. Homogeneous hypersurfaces are precisely the codimension 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)]

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)]

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)]

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)]

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 equidistant hypersurfaces have constant mean curvature. Equivalently, if M is a regular level set of a function f : U

• pen

⊂ ¯ M → R such that |∇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 equidistant hypersurfaces have constant mean curvature. Equivalently, if M is a regular level set of a function f : U

• pen

⊂ ¯ M → R such that |∇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 equidistant hypersurfaces have constant mean curvature. Equivalently, if M is a regular level set of a function f : U

• pen

⊂ ¯ M → R such that |∇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 equidistant hypersurfaces have constant mean curvature. Equivalently, if M is a regular level set of a function f : U

• pen

⊂ ¯ M → R such that |∇f | = a ◦ f and ∆f = b ◦ f , for smooth functions a, b. M homogeneous hypersurface ⇓ M isoparametric hypersurface with 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 hyperplanes Rn−1 Concentric spheres Sn−1 Generalized cylinders Sk × 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 equidistant hypersurfaces Tubes around a

• tot. 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, M¨ unzner (1981)] All isoparametric hypersurfaces are homogeneous or of FKM-type [Cartan; M¨ unzner; 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)]

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

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

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

There are countably many inhomogeneous examples, all of them with nonconstant 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 = 15 [DV (2016)] and in HPn, n = 7 [DV, Gorodski (2018)]

There are countably many inhomogeneous examples, all of them with nonconstant 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 hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M.

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global 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 geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M. Symmetric spaces are complete and homogeneous ¯ M ∼ = G/K, where G = Isom0( ¯ M) and K = {g ∈ G : g(o) = o} are Lie groups, and o ∈ ¯ M is a base point

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M. Symmetric spaces are complete and homogeneous ¯ M ∼ = G/K, where G = Isom0( ¯ M) and K = {g ∈ G : g(o) = o} are Lie groups, and o ∈ ¯ M is a base point compact type noncompact type Euclidean type duality

Symmetric spaces of noncompact type

Definition [Cartan (1926)]

A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σp : expp(v) → expp(−v), v ∈ Tp ¯ M, around each p ∈ ¯ M is a global isometry of ¯ M. Symmetric spaces are complete and homogeneous ¯ M ∼ = G/K, where G = Isom0( ¯ M) and K = {g ∈ G : g(o) = o} are Lie groups, and o ∈ ¯ M is a base point compact type noncompact type Euclidean type duality

¯ 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

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Iwasawa decomposition

g = k ⊕ a ⊕ n n is nilpotent

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Iwasawa decomposition

g = k ⊕ a ⊕ n n is nilpotent G

diffeo.

∼ = K × A × N

K A N

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Iwasawa decomposition

g = k ⊕ a ⊕ n n is nilpotent G

diffeo.

∼ = K × A × N

K A N

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

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Iwasawa decomposition

g = k ⊕ a ⊕ n n is nilpotent G

diffeo.

∼ = 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

Symmetric spaces of noncompact type

¯ M ∼ = G/K symmetric space of noncompact type = ⇒ ¯ M

diffeo.

∼ = Bn g = k ⊕ p Cartan decomposition, p ∼ = To ¯ M a maximal abelian subspace of p, rank ¯ M := dim a

Iwasawa decomposition

g = k ⊕ a ⊕ n n is nilpotent G

diffeo.

∼ = 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/K

isom.

∼ = AN symmetric space of noncompact type, rank ¯ M = 1

Symmetric spaces of noncompact type and rank one

¯ M ∼ = G/K

isom.

∼ = 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 one

¯ M ∼ = G/K

isom.

∼ = 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

¯ M RHn CHn HHn OH2

SO0(1,n) SO(n) SU(1,n) S(U(1)×U(n)) Sp(1,n) Sp(1)×Sp(n) F−20

4

Spin(9)

v Rn−1 Cn−1 Hn−1 O dim z 1 3 7

Symmetric spaces of noncompact type

1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

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, Br¨ uck (2001)]

Tubes around totally geodesic submanifolds P in FHn are homogeneous if and 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/K

isom.

∼ = 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/K

isom.

∼ = 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, Br¨ uck (2001), Berndt, Tamaru (2003)]

Orbit equivalent to the action of: N ❀ horosphere foliation The connected subgroup of G with Lie algebra a ⊕ w ⊕ z, where w is a (real) hyperplane in v

Cohomogeneity one actions on hyperbolic spaces

FHn ∼ = G/K

isom.

∼ = 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, Br¨ uck (2001)]

w v (real) subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw

Cohomogeneity one actions on hyperbolic spaces

FHn ∼ = G/K

isom.

∼ = 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, Br¨ uck (2001)]

w v (real) subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw The tubes around Sw are homogeneous if and only if NK0(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/K

isom.

∼ = 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, Br¨ uck (2001)]

w v (real) subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw The tubes around Sw are homogeneous if and only if NK0(w) acts transitively on the unit sphere of w⊥ (the orthogonal complement of w in v)

Sw

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n), K0 = NK(a)

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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.

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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.

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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.

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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.

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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 such that NK0(w) acts transitively on the unit sphere of w⊥.

Cohomogeneity one actions on hyperbolic spaces

FHn

isom.

∼ = 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 such that 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

FHn

isom.

∼ = 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 such that 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 hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw Sw is a homogeneous minimal submanifold

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra Sw connected subgroup of AN with Lie algebra sw Sw is a homogeneous minimal submanifold The tubes around Sw are isoparametric

Sw tube

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra 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

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra 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 isoparametric families of hypersurfaces with nonconstant principal curvatures

New isoparametric hypersurfaces

FHn

isom.

∼ = AN symmetric space of noncompact type and rank one a ⊕ n = a ⊕ v ⊕ z, a ∼ = R, z = Z(n)

New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)]

w v real subspace = ⇒ sw = a ⊕ w ⊕ z is a Lie algebra 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 isoparametric families of hypersurfaces with nonconstant principal curvatures In OH2 there is one inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures (when dim w = 3)

Classification in the complex hyperbolic space

Theorem [D´ ıaz-Ramos, DV, Sanmart´ ın-L´

• pez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn is isoparametric 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 [D´ ıaz-Ramos, DV, Sanmart´ ın-L´

• pez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn is isoparametric 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 [D´ ıaz-Ramos, DV, Sanmart´ ın-L´

• pez (2017)]

A connected hypersurface M in the complex hyperbolic space CHn is isoparametric 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, Br¨ uck (2001)] and inhomogeneous [D´ ıaz-Ramos, DV (2012)] examples, depending on w ⊂ v

The quaternionic hyperbolic space

1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one 1

Cohomogeneity one actions

2

Isoparametric hypersurfaces

3 The quaternionic hyperbolic space

The quaternionic hyperbolic space

Problem

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

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 on the unit sphere of w⊥.

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 on the unit sphere of w⊥. K0 ∼ = Sp(n)Sp(1) acts on v ∼ = Hn via (A, q) · v = Avq−1

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 on the 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 .

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 on the 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

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

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

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

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 K¨ ahler 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 K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ {0} with respect to V is ϕ.

Getting intuition in the complex setting

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

Definition

The K¨ ahler 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 K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ {0} with respect to V is ϕ. Totally real subspaces have constant K¨ ahler angle π/2 Complex subspaces have constant K¨ ahler angle 0 V = spanR{e1, Je1, e2} does not have constant K¨ ahler angle

Getting intuition in the complex setting

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

Definition

The K¨ ahler 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 K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ {0} with respect to V is ϕ. Totally real subspaces have constant K¨ ahler angle π/2 Complex subspaces have constant K¨ ahler angle 0 V = spanR{e1, Je1, e2} does not have constant K¨ ahler angle

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant K¨ ahler angle.

Getting intuition in the complex setting

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

Definition

The K¨ ahler 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 K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ {0} with respect to V is ϕ. Totally real subspaces have constant K¨ ahler angle π/2 Complex subspaces have constant K¨ ahler angle 0 V = spanR{e1, Je1, e2} does not have constant K¨ ahler angle

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Cn is protohomogeneous if and only if it has constant K¨ ahler angle. Moreover, V has constant K¨ ahler angle ϕ ∈ [0, π/2) if and only if V = span{e1, cos ϕJe1 + sin ϕJe2, . . . , ek, cos ϕJe2k−1 + sin ϕJe2k}.

Back to the quaternionic setting

Problem

Classify protohomogeneous subspaces of Hn.

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: J2

i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

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: J2

i = −Id and JiJi+1 = Ji+2 = −Ji+1Ji

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

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: J2

i = −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 K¨ ahler angle of v with respect to V is the triple (ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv are cos2 ϕi v, v, i = 1, 2, 3.

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: J2

i = −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 K¨ ahler angle of v with respect to V is the triple (ϕ1, ϕ2, ϕ3), with ϕ1 ≤ ϕ2 ≤ ϕ3, such that the eigenvalues of Lv are cos2 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, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle.

Back to the quaternionic setting

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle. There are subspaces V with constant quaternionic K¨ ahler 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)...

Back to the quaternionic setting

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle. There are subspaces V with constant quaternionic K¨ ahler 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 K¨ ahler angle of a subspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Back to the quaternionic setting

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle. There are subspaces V with constant quaternionic K¨ ahler 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 K¨ ahler angle of a subspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic K¨ ahler angle.

Back to the quaternionic setting

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle. There are subspaces V with constant quaternionic K¨ ahler 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 K¨ ahler angle of a subspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic K¨ ahler angle.

Question

Does constant quaternionic K¨ ahler angle imply protohomogeneous?

Back to the quaternionic setting

Proposition [Berndt, Br¨ uck (2001)]

V ⊂ Hn protohomogeneous ⇒ V has constant quaternionic K¨ ahler angle. There are subspaces V with constant quaternionic K¨ ahler 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 K¨ ahler angle of a subspace V , e.g. (0, 0, ϕ), ϕ ∈ (0, π/2)

Problem

Classify real subspaces of Hn with constant quaternionic K¨ ahler angle.

Question

Does constant quaternionic K¨ ahler angle imply protohomogeneous?

Theorem [D´ ıaz-Ramos, DV (2013))]

The tubes around Sw have constant principal curvatures if and only if w⊥ ⊂ v has constant quaternionic K¨ ahler angle.

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3)

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3) Sk−1 unit sphere of V ,

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler angle Φ(V ) = (ϕ1, ϕ2, ϕ3) Sk−1 unit sphere of V , π: Hn → V orthogonal projection onto V

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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}

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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).

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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].

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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].

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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). Case k ≡ 0 (mod 4).

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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). Case k ≡ 0 (mod 4).

Protohomogeneous subspaces in Hn

V protohomogeneous real subspace of Hn, dim V = k ⇒ V constant quaternionic K¨ ahler 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). Case k ≡ 0 (mod 4). ?

Protohomogeneous subspaces in Hn

Problem

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

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3)

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5.

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji)

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

2 Define Pi =

1 cos ϕi πJi : V → V .

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

2 Define Pi =

1 cos ϕi πJi : V → V . Then PiPj + PjPi = −2δijId.

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

2 Define Pi =

1 cos ϕi πJi : V → V . Then PiPj + PjPi = −2δijId.

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

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

2 Define Pi =

1 cos ϕi πJi : V → V . Then PiPj + PjPi = −2δijId.

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, dim V± = 4.

Protohomogeneous subspaces in Hn

Problem

Classify protohomogeneous real subspaces V ⊂ Hn with dim V = k = 4r V protohomogeneous subspace of Hn, dim V = 4r, Φ(V ) = (ϕ1, ϕ2, ϕ3) Assume k ≥ 5. For simplicity, assume ϕ3 = π/2.

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

i = −Id,

JiJi+1 = Ji+2 = −Ji+1Ji) such that the K¨ ahler angle of any v ∈ Sk−1 with respect to V and the complex structure Ji is ϕi.

2 Define Pi =

1 cos ϕi πJi : V → V . Then PiPj + PjPi = −2δijId.

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, dim V± = 4.

5 Each factor has constant quaternionic K¨

ahler angle (ϕ1, ϕ2, ϕ3).

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

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

V+, which exists if and only if cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1. V−, which exists if and only if cos ϕ1 + cos ϕ2 − cos ϕ3 ≤ 1.

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

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

V+, which exists if and only if cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1. V−, which exists if and only if cos ϕ1 + cos ϕ2 − cos ϕ3 ≤ 1.

7 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−.

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

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

V+, which exists if and only if cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1. V−, which exists if and only if cos ϕ1 + cos ϕ2 − cos ϕ3 ≤ 1.

7 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−. 8 If V , with dim V = 4r, then either V =

• V+ or V =
• V−.

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

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

V+, which exists if and only if cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1. V−, which exists if and only if cos ϕ1 + cos ϕ2 − cos ϕ3 ≤ 1.

7 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−. 8 If V , with dim V = 4r, then either V =

• V+ or V =
• V−.

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

Protohomogeneous subspaces in Hn

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

• V+
• ⊕
• V−
• V+ and V− two inequivalent irreducible Cl(3)-modules

dim V± = 4, Φ(V±) = (ϕ1, ϕ2, ϕ3)

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

V+, which exists if and only if cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1. V−, which exists if and only if cos ϕ1 + cos ϕ2 − cos ϕ3 ≤ 1.

7 ∃T ∈ Sp(n)Sp(1) such that TV+ = V−. 8 If V , with dim V = 4r, then either V =

• V+ or V =
• V−.

From this, one can obtain the classification of protohomogeneous subspaces 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, dim V± = 4

New isoparametric hypersurfaces

V = r+

• V+
• ⊕

r−

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

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1

New isoparametric hypersurfaces

V = r+

• V+
• ⊕

r−

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

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1

Theorem [D´ ıaz-Ramos, DV, Rodr´ ıguez-V´ azquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn with constant quaternionic K¨ ahler angle.

New isoparametric hypersurfaces

V = r+

• V+
• ⊕

r−

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

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1

Theorem [D´ ıaz-Ramos, DV, Rodr´ ıguez-V´ azquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn with constant quaternionic K¨ ahler angle. HHn+1

isom.

∼ = AN, a ⊕ n = a ⊕ v ⊕ z, v ∼ = Hn

New isoparametric hypersurfaces

V = r+

• V+
• ⊕

r−

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

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1

Theorem [D´ ıaz-Ramos, DV, Rodr´ ıguez-V´ azquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn with constant quaternionic K¨ ahler angle. HHn+1

isom.

∼ = AN, a ⊕ n = a ⊕ v ⊕ z, v ∼ = Hn w := orthogonal complement of V in v sw = a ⊕ w ⊕ z ❀ Sw connected subgroup of AN

New isoparametric hypersurfaces

V = r+

• V+
• ⊕

r−

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

Φ(V±) = (ϕ1, ϕ2, ϕ3) with cos ϕ1 + cos ϕ2 + cos ϕ3 ≤ 1

Theorem [D´ ıaz-Ramos, DV, Rodr´ ıguez-V´ azquez (2019)]

If r+, r− ≥ 1, then V is a non-protohomogeneous subspace of Hn with constant quaternionic K¨ ahler angle. HHn+1

isom.

∼ = AN, a ⊕ n = a ⊕ v ⊕ z, v ∼ = Hn w := orthogonal complement of V in v sw = a ⊕ w ⊕ z ❀ Sw connected subgroup of AN

Theorem [D´ ıaz-Ramos, DV, Rodr´ ıguez-V´ azquez (2019)]

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