. Gauss map of real hypersurfaces in non-flat complex space forms and twistor space of complex 2 -plane Grassmannian . Makoto Kimura(Ibaraki University) June 5, 2019 Workshop on the isoparametric theory Beijing Normal University Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Contents Gauss map of hypersurface in S n +1 , and parallel hypersurfaces, Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Contents Gauss map of hypersurface in S n +1 , and parallel hypersurfaces, Gauss map of real hypersurfaces in CP n and ahler structure of G 2 ( C n +1 ) , quaternionic K¨ Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Contents Gauss map of hypersurface in S n +1 , and parallel hypersurfaces, Gauss map of real hypersurfaces in CP n and ahler structure of G 2 ( C n +1 ) , quaternionic K¨ Hopf hypersurfaces in CH n and para-quaternionic K¨ ahler structure of G 1 , 1 ( C n +1 ) , 1 Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Contents Gauss map of hypersurface in S n +1 , and parallel hypersurfaces, Gauss map of real hypersurfaces in CP n and ahler structure of G 2 ( C n +1 ) , quaternionic K¨ Hopf hypersurfaces in CH n and para-quaternionic K¨ ahler structure of G 1 , 1 ( C n +1 ) , 1 Ruled Lagrangian submanifolds in CP n and some quarter dimensional submanifolds of G 2 ( C n +1 ) . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of hypersurfaces in sphere For an immersion x : M n → S n +1 ⊂ R n +2 , let x ( p ) ∈ S n +1 ⊂ R n +2 be the position vector at p ∈ M , and let N p be a unit normal vector of oriented hypersurface M ⊂ S n +1 at p ∈ M . G 2 ( R n +2 ) ∼ Then the Gauss map γ : M → � = Q n is defined by γ ( p ) = x ( p ) ∧ N p (B. Palmer, 1997, Diff. Geom. Appl.). Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of hypersurfaces in sphere Then the image of the Gauss map γ ( M ) is a Lagrangian submanifold in complex quadric Q n . Moreover, if M n ⊂ S n +1 is either isoparametric or austere, then γ ( M ) ⊂ Q n is a minimal Lagrangian submanifold. Also for each parallel hypersurface M r := cos rx + sin rN ( r ∈ R ) of M , the Gauss image is not changed: γ ( M ) = γ ( M r ) . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of hypersurfaces in sphere Conversely, let γ : M n → Q n = � G 2 ( R n +1 ) be a Lagrangian immersion. Then we have a lift γ : M n → V 2 ( R n +2 ) to real Stiefel manifold ˜ (w.r.t. the fibration V 2 ( R n +1 ) → � G 2 ( R n +1 ) ) , and with respect to a contact (Sasakian) structure of V 2 ( R n +1 ) , ˜ γ is a Legendrian immersion. If we γ ( p ) = ( u 1 ( p ) , u 2 ( p )) ∈ V 2 ( R n +2 ) , then denote ˜ for r ∈ R , p �→ cos ru 1 ( p ) + sin ru 2 ( p ) gives original family of “parallel hypersurfaces” in S n +1 . Anciaux (2014, Trans. Amer. Math. Soc.) generalized the result to hypersurfaces in hyperbolic space and indefinite real space forms. Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of real hypersurface in CP n For a real hypersurface M 2 n − 1 in CP n , we consider the following diagram: S 2 n +1 − π − 1 ( M ) − C n +1 − → − → w ι . � � π M 2 n − 1 CP n − − → x For p ∈ M , take a point z p ∈ π − 1 ( x ( p )) ⊂ π − 1 ( M ) and let N ′ p be a horizontal lift of unit normal of M ⊂ CP n at z p . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of real hypersurface in CP n If we put γ ( p ) = span C { z p , N ′ p } , then the map γ : M → G 2 ( C n +1 ) is well-defined. We call γ as the Gauss map of real hypersurface M in CP n . Note that for a parallel hypersurface M r := π (cos rz p + sin rN ′ p ) of M , the image of the Gauss map γ : M 2 n − 1 → CP n is not changed: γ ( M ) = γ ( M r ) . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Hopf hypersurfaces in K¨ ahler manifold For a real hypersurface M 2 n − 1 in K¨ ahler manifold ( � M n , J ) and a unit normal vector N ,a vector ξ := − JN tangent to M is called the structure vector of M . And when ξ is an eigenvector of the shape operator A of M , i.e., Aξ = µξ , we call M a Hopf hypersurface in � M , and µ the Hopf (principal) curvature. Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Hopf hypersurfaces in K¨ ahler manifold If � M is a non-flat complex space form � M n ( c ) ( c ̸ = 0) , then µ is constant on M (Y. Maeda and Ki-Suh) and when c > 0 , each integral curve of ξ is a geodesic (resp. equidistance curve from a geodesic) in CP 1 ⊂ CP n , provided µ = 0 (resp. µ ̸ = 0 ). They form concentric circles in CP 1 = S 2 . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Hopf hypersurfaces in complex projective space A real hypersurface which lies on a tube over a complex submanifold Σ in CP n is Hopf. Conversely, if a Hopf hypersurface M in CP n (4) satisfies Aξ = µξ , and for r ∈ (0 , π/ 2) with µ = 2 cot 2 r , r ∈ (0 , π/ 2) , if the rank of the focal map φ r : M → CP n is constant, then φ r ( M ) is a complex submanifold of CP n (4) and M lies on a tube over φ r ( M ) . (Cecil-Ryan, 1982, Trans. Amer. Math. Soc.). Also they showed that if M is a Hopf hypersurface in CP n , then each parallel hypersurface M r is also Hopf. Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Hopf hypersurfaces in complex projective space Typical example of Hopf hypersurface is a geodesic hypersphere. R. Takagi (Osaka J.M. ’70) classified all homogeneous real hypersurfaces in CP n , and they are obtained as a regular orbit of isotropy representation of Hermitian symmetric space of rank 2 . Also he showed that they are all Hopf. We know that . K., Trans. AMS, ’86 . . a Hopf hypersurface M in CP n has constant principal curvatures if and only if M is a homogeneous real hypersurface. . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Hopf hypersurfaces in complex projective space After that, Borisenko(2001, Illinois J. Math.) obtained some results concerning Hopf hypersurfaces in CP n without assumption of rank about the focal map. For example, he showed that compact embedded Hopf hypersurface in CP n lies on a tube over an algebraic variety. We will give a characterization of Hopf hypersurface M in CP n by using the Gauss map. γ : M → G 2 ( C n +1 ) . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Quaternionic K¨ ahler manifold Complex 2-plane Grassmann manifold � M = G 2 ( C n +1 ) has two important geometric structures, (i) K¨ ahler and (ii) quaternionic K¨ ahler structure (˜ g, Q ) : Here, ˜ g is a Riemannian metric of � M , Q is a subbundle of End T � M with rank 3 , satisfying: For each p ∈ � M , there exists a neighborhood U ∋ p , such that there exists local frame field { ˜ I 1 , ˜ I 2 , ˜ I 3 } of Q . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Quaternionic K¨ ahler manifold ˜ 1 = ˜ 2 = ˜ I 1 ˜ ˜ I 2 = − ˜ I 2 ˜ I 1 = ˜ I 2 I 2 I 2 3 = − 1 , I 3 , I 2 ˜ ˜ I 3 = − ˜ I 3 ˜ I 2 = ˜ I 3 ˜ ˜ I 1 = − ˜ I 1 ˜ I 3 = ˜ I 1 , I 2 . For each L ∈ Q p , ˜ g is invariant, i.e., ˜ g p ( LX, Y ) + ˜ g p ( X, LY ) = 0 for X, Y ∈ T p � M , p ∈ � M . Vector bundle Q is parallel with respect to the Levi-Civita connection of g at End T � ˜ M . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Almost Hermitian submanifolds in Q.K. manifold A submanifold M 2 m in quaternionic K¨ ahler manifold � M is called almost Hermitian submanifold, if there exists a section ˜ I of vector I 2 = − 1 , and bundle Q | M over M such that (1) ˜ (2) ˜ IT M = T M .if we write the almost complex structure on M which is induced by ˜ I as I , then with respect to the induced metric, ( M, I ) is an almost Hermitian manifold. Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Totally complex submanifold of Q.K. manifold In particular, when almost Hermitian submanifold ( M, ¯ g, I ) is K¨ ahler, we call M a K¨ ahler ahler manifold � submanifold of quaternionic K¨ M . Similarly, an almost Hermitian submanifold ( M, ¯ g, I ) is called totally complex submanifold if at each point p ∈ M , with respect to ˜ L ∈ Q p which anti-commute with ˜ I p , ˜ LT p M ⊥ T p M hold. In quaternionic K¨ ahler manifold, a submanifold is totally complex if and only if it is K¨ ahler (Alekseevsky-Marchiafava, 2001, Osaka J. Math.). Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Gauss map of real hypersurface in CP n . Theorem 1. (K., Diff. Geom. Appl. 2014) . . Let M 2 n − 1 be a real hypersurface in complex projective space CP n , and let γ : M → G 2 ( C n +1 ) be the Gauss map. If M is not Hopf, then the Gauss map γ is an immersion. If M is a Hopf hypersurface, then the image γ ( M ) is a half-dimensional totally complex (hence minimal) submanifold of G 2 ( C n +1 ) . And any Hopf hypersurface M in CP n is a total space of a circle bundle over a K¨ ahler manifold such that the fibration is nothing but the Gauss map γ : M → γ ( M ) ⊂ G 2 ( C n +1 ) . . Makoto Kimura(Ibaraki University) Gauss map of real hypersurfaces
Recommend
More recommend
