Isoparametric submanifolds admitting a reflective focal submanifold - PowerPoint PPT Presentation

. Isoparametric submanifolds admitting a reflective focal submanifold in symmetirc spaces of non-compact type . Naoyuki Koike Tokyo University of Science koike@rs.kagu.tus.ac.jp Workshop on the Isoparametric Theory Beijing Normal University June 3, 2019

Content 1. Introduction 2. Isoparametric submanifold and complex equifocal submanifold 3. ∞ -dimensional isoparametric submanifold submanifold 4. ∞ -dim. anti-Kaehler isoparametric submanifold 5. Outline of the proofs of results

1. Introduction

Lift to Hilbert space � M := ( π ◦ φ ) − 1 ( M ) H 0 ([0 , 1] , g ) ⊂ φ lift G π M ⊂ G/K G/K : a simply connected symmetric space of compact type . Theorem A.1(Terng-Thorbergsson, 1995) . � M : equifocal ⇐ ⇒ M : isoparametric .

Homogeneity of ∞ -dim. isoparametric submanifolds • In 1999, Heintze-Liu proved the homogeneity theorem for isoparametric submanifolds in a Hilbert space. • In 2002, Christ proved the homogeneity theorem for an equifocal submanifold M in G/K by applying Heintze-Liu’ theorem to � M = ( π ◦ φ ) − 1 ( M ) ⊂ H 0 ([0 , 1] , g ) . In the proof, he used the fact that � M is homogeneous by a Banach Lie group action. • In 2012, Gorodski-Heintze proved that the homogeneity in Heintze-Liu’s theorem means the homogeneity by a Banach Lie group action.

Homogeneity of ∞ -dim. isoparametric submanifolds V : (seprable) Hilbert space M ( ⊂ V ) : complete proper Fredholm submanifold . Theorem A.2(Heintze-Liu, 1999) . M : full irreducible isoparametric submanifold of codim M ≥ 2 in V = ⇒ M : homogeneous (i.e., M = H · p ( ∃ H ⊂ I ( V ) )) . Remark H is given by H := { F ∈ I ( V ) | F ( M ) = M } . I ( V ) is not a Banach Lie group. Hence H also is not a Banach Lie group in general.

A Banach Lie group of isometries  �  � ∃ { F t } t ∈ [0 , 1] : a one para transf . gr .    �      �   • F 1 = F �  I b ( V ) := F ∈ I ( V ) �  � s . t . • the Killing vec . fd . ass . to     �      � { F t } t ∈ [0 , 1] is defined on V V X u u I ( V ) X F I ( V ) id t �→ F t ( u ) t �→ F t X : the ass. vec. field of { F t } t ∈ [0 , 1]

A Banach Lie group of isometries . Fact. . I b ( V ) is a Banach Lie group. . Proof ϕ : I b ( V ) − → o b ( V ) ⊕ V (Banach space) ( dA t � � ) � � , db t � � F �→ � � dt dt t =0 t =0 ( F t ( u ) = A t ( u ) + b t ( F 1 = F )) D := { ( L F ( U ) , ( ϕ ◦ L − 1 F ) | L F ( U ) ) } F ∈ I b ( V ) gives a Banach Lie group str. of I b ( V ) , where U is a suff. small nbd of id . � � � � X u = dA t u + db t � � ( u ∈ V ) Remark � � dt dt t =0 t =0

An example of an element of I b ( V ) { } ∞ ∑ V := l ∞ = ( a i ) ∞ a 2 Example 1 i =1 | i < ∞ i =1 ( u ∈ V ) F t ( u ) := A t ( u ) + b t ( ( ) ) cos t − sin t ∞ k k A t = ⊕ sin t cos t k =1 k k Then the Killing vec. fd. X ass. to { F t } t is given by X u = B ( u ) + b ( u ∈ V ) ( ( ) ) − 1 0 ∞ k ⊕ B = 1 0 k =1 k Since B is bounded, X is defined on V . Hence we have F 1 ∈ I b ( V ) ．

An example of an element of I ( V ) \ I b ( V ) V := l ∞ Example 2 ( u ∈ V ) F t ( u ) := A t ( u ) + b t ( ( ) ) ∞ cos kt − sin kt A t = ⊕ sin kt cos kt k =1 Then the Killing vec. fd. X ass. to { F t } t is given by X u = Bu + b ( u ∈ V ) ( ( ) ) 0 − k ∞ B = ⊕ k 0 k =1 Since B is not bounded, X is not defined on V . Hence we have F 1 / ∈ I ( V ) \ I b ( V ) .

An example of an element of I ( V ) \ I b ( V )   1   = (1 , 1 , 2 , 2 , 3 , 3 , · · · ) ∈ l ∞ [ ] u = i +1 2 ∈ l ∞ B ( u ) = ( − 1 , 1 , − 1 , 1 , − 1 , 1 , · · · ) / Hence X is not defined at u .

Homogeneity of ∞ -dim. isoparametric submanifolds M ( ⊂ V ) : complete proper Fredholm submanifold . Theorem A.3(Gorodski-Heintze, 2012) . M : full irreducible isoparametric submanifold of codim M ≥ 2 in V = ⇒ M = H b · p ( ∃ H b ⊂ I b ( V ) ) . Remark H b = { F ∈ I b ( V ) | F ( M ) = M }

Homogeneity of equifocal submanifolds G/K : simply connected symmetric space of compact type . Theorem A.4(Christ) . M : full irreducible equifocal submanifold of codim M ≥ 2 in G/K = ⇒ M : homogeneous (i.e., M = H · p ( ∃ H ⊂ I ( G/K ) )) .

Complexification and Lift to ∞ -dim. anti-Kaheler space M C := ( π ◦ φ ) − 1 ( M C ) ⊂ H 0 ([0 , 1] , g C ) � φ G C lift π M C ⊂ G C /K C M ⊂ G/K extrinsic complexification G/K : symmetric space of non-compact type . Theorem B.1(K, 2005) . M : complex equifocal � M C : anti-Kaehler isoparametric ⇐ ⇒ .

Homogeneity of ∞ -dim. anti-Kaehler isoparametric submanifolds V : ∞ -dim. anti-Kaehler space M ( ⊂ V ) : complete anti-Kaehler Fredholm submanifold . Theorem B.2(K,2014) . M : full irr. anti-Kaehler isoparametric submanifold of codim M ≥ 2 with J -diagonalizable shape op. in V = ⇒ M : homogeneous (i.e., M = H · p ( ∃ H ⊂ I ( V ) )) . Remark H is given by H := { F ∈ I ( V ) | F ( M ) = M } . I ( V ) is not a Banach Lie group. Hence H also is not a Banach Lie group in general.

Homogeneity of ∞ -dim. anti-Kaehler isoparametric submanifolds  �  �  ∃ { F t } t ∈ [0 , 1] : a one para transf . gr .   �      �   • F 1 = F �  F ∈ I ( V ) I b ( V ) := �  � s . t .  • the hol . Killing v . fd . ass . to    �      � { F t } t ∈ [0 , 1] is defined on V . Theorem B.3(K,2017) . M : full irr. anti-Kaehler isoparametric submanifold of codim M ≥ 2 with J -diagonalizable shape op. in V = ⇒ M : homogeneous (i.e., M = H b · p ( ∃ H b ⊂ I b ( V ) )) . Remark H b = { F ∈ I b ( V ) | F ( M ) = M }

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type G/K : symmetric space of non-compact type ( ∗ C ) For any unit normal vec. v of M , the nullity spaces of the complex focal radii along the normal geodesic γ v span ( T p M ) C ∩ ((Ker A v ∩ Ker R ( v )) C ) ⊥ . . Theorem B.4(K,2018) . M : full irreducible curvature-adapted isoparametric C ω -submanifold of codim M ≥ 2 in G/K s.t. ( ∗ C ) ⇒ M : homogeneous (i.e., M = H · p ( ∃ H ⊂ I ( G/K ) )) = . ( M : curv.-adapted & ( ∗ C ) ⇒ � M C : has J -diag. shape op.)

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type . Theorem B.5(K,2018) . M : full irreducible isoparametric C ω -submanifold of codim M ≥ 2 in G/K admitting a reflective focal submanifold = ⇒ M : a principal orbit of a Hermann type action . Remark (i) Let H be a symmetric subgroup of G . Then the natural action of H on G/K is called a Hermann type action. (ii) Principal orbits of a Hermann type action are curvature-adapted isoparametric submanifolds.

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type . Question . Can we delete the assumption of the real anlayticity . of M in Theorem B.4? ( ∗ R ) For any unit normal vec. v of M , the nullity spaces of the focal radii along γ v span T p M . . Theorem B.6(K,2018) . M : full irreducible curvature-adapted isoparametric C ∞ -submanifold of codim M ≥ 3 in G/K s.t. ( ∗ R ) = ⇒ M : homogeneous (i.e., M = H · p ( ∃ H ⊂ I ( G/K ) )) ( M : a principal orbit of the isotropy action of G/K ) .

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type We proved this theorem by constructing a Tits building associtaed to M and using Burns-Spatzier’s theorem (1987).

2. Isoparametric submanifold and complex equifocal submanifold

Equifocal submanifold G/K : symmetric space of compact type M ( ⊂ G/K ) : compact submanifold . Def(Equifocal submanifold) . M : equifocal submanifold   • M is a submanifold with flat section      • The normal holonomy gr. of M is trivial  ⇐ ⇒ • For each parallel normal vec. fd. � v ,  def   the focal radii along γ v p is independent     of p ∈ M .

Isoparametric submanifold ( � M, � g ) : complete Riemannian manifold M ( ⊂ � M ) : complete submanifold . Def(Isoparametric submanifold in Heintze-Liu-Olmos-sense) . M : isoparametric submanifold with flat section  • M is a submanifold with flat section     • The normal holonomy gr. of M is trivial ⇐ ⇒  • Sufficciently close parallel submanifolds of M def    are of CMC w.r.t. the radial direction . In this talk, we call this submanifold “isoparametic submanifold” for simplicity.

Isoparametric submanifold M v � η � v ( M ) the radial directions the section of M thr. p p

Equifocality and isoparametricness . Proposition 2.1(Heintze-Liu-Olmos,2006). . Assume that M is compact. Then M : equifocal ⇐ ⇒ M : isoparametric .

Complex focal radius G/K : symmetirc space of non-compact type G C /K C : the complexification of G/K M ( ⊂ G/K ) : C ω -submanifold in G/K M C ( ⊂ G C /K C ) : the complexification of M ( ⊂ G/K ) γ v : the normal geodesic of M of direction v ( ∈ T ⊥ p M ) ( � v � = 1 ) γ C v : the complexification of γ v

Complex focal radius . Def(Complex focal radius) . √− 1 : complex focal radius along γ v z 0 = s 0 + t 0 γ C ( z 0 ) : a focal point of M C along s �→ γ C ( sz 0 ) ⇐ ⇒ . def