Isometric Reeb Flow and Related Results on Hermitian Symmetric Spaces of Rank 2

28/08/2013
Auteurs : Young Jin Suh
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Isometric Reeb Flow and Related Results on Hermitian Symmetric Spaces of Rank 2

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Isometric Reeb Flow and Related Results on Hermitian Symmetric Spaces of Rank 2 Young Jin Suh Department of Mathematics Kyungpook National University Taegu 702-701, Korea Ecole des Meines, Paris, France Geometric Science of Information, GSI’13 28-30th, August, 2013 E-mail: yjsuh@knu.ac.kr August 29, 2013 Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Contents 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Contents 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Contents 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Hermitian Symmetric Spaces Hereafter let us note that HSSP means Hermitian Symmetric Space. HSSP of compact type with rank 1: CPm, QPm HSSP of noncompact type with rank 1: CHm, QHm. HSSP of compact type with rank 2: SU(2 + q)/S(U(2)×U(q)), Qm, SO(8)/U(4), Sp(2)/U(2) and (e6(−78), SO(10) + R) HSSP of compact type with rank 2: SU(2, q)/S(U(2)×U(q)), Q∗m, SO∗(8)/U(4), Sp(2, R)/U(2) and (e6(2), SO(10) + R) (See Helgason [6], [7]). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Hermitian Symmetric Spaces Hereafter let us note that HSSP means Hermitian Symmetric Space. HSSP of compact type with rank 1: CPm, QPm HSSP of noncompact type with rank 1: CHm, QHm. HSSP of compact type with rank 2: SU(2 + q)/S(U(2)×U(q)), Qm, SO(8)/U(4), Sp(2)/U(2) and (e6(−78), SO(10) + R) HSSP of compact type with rank 2: SU(2, q)/S(U(2)×U(q)), Q∗m, SO∗(8)/U(4), Sp(2, R)/U(2) and (e6(2), SO(10) + R) (See Helgason [6], [7]). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Hermitian Symmetric Spaces Hereafter let us note that HSSP means Hermitian Symmetric Space. HSSP of compact type with rank 1: CPm, QPm HSSP of noncompact type with rank 1: CHm, QHm. HSSP of compact type with rank 2: SU(2 + q)/S(U(2)×U(q)), Qm, SO(8)/U(4), Sp(2)/U(2) and (e6(−78), SO(10) + R) HSSP of compact type with rank 2: SU(2, q)/S(U(2)×U(q)), Q∗m, SO∗(8)/U(4), Sp(2, R)/U(2) and (e6(2), SO(10) + R) (See Helgason [6], [7]). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Hermitian Symmetric Spaces Hereafter let us note that HSSP means Hermitian Symmetric Space. HSSP of compact type with rank 1: CPm, QPm HSSP of noncompact type with rank 1: CHm, QHm. HSSP of compact type with rank 2: SU(2 + q)/S(U(2)×U(q)), Qm, SO(8)/U(4), Sp(2)/U(2) and (e6(−78), SO(10) + R) HSSP of compact type with rank 2: SU(2, q)/S(U(2)×U(q)), Q∗m, SO∗(8)/U(4), Sp(2, R)/U(2) and (e6(2), SO(10) + R) (See Helgason [6], [7]). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Hypersurfaces in Hermitian Symmetric Spaces Let M be a hypersurfaces in a Hermitian Symmetric Space ¯M with Kaehler structure J. AX = − ¯ X N : Weingarten formula Here A: the shape operator of M in ¯M. ξ = −JN : the Reeb vector field. JX = φX + η(X)N, X ξ = φAX for any vector field X∈Γ(M). Then (φ, ξ, η, g): almost contact structure on a hypersurface M Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Define) A hypersurfcae M: Isometric Reeb Flow ⇐⇒ Lξg = 0 ⇐⇒ g(dφt X, dφt Y) = g(X, Y) for any X, Y∈Γ(M), where φt denotes a one parameter group, which is said to be an isometric Reeb flow of M, defined by dφt dt = ξ(φt (p)), φ0(p) = p, ˙φ0(p) = ξ(p). Note) Lξg = 0 ⇐⇒ jξi + iξj = 0, ξ: skew-symmetric ⇐⇒ g( X ξ, Y) + g( Y ξ, X) = 0 ⇐⇒ g((φA − Aφ)X, Y) = 0 for any X, Y∈Γ(M). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Define) A hypersurfcae M: Isometric Reeb Flow ⇐⇒ Lξg = 0 ⇐⇒ g(dφt X, dφt Y) = g(X, Y) for any X, Y∈Γ(M), where φt denotes a one parameter group, which is said to be an isometric Reeb flow of M, defined by dφt dt = ξ(φt (p)), φ0(p) = p, ˙φ0(p) = ξ(p). Note) Lξg = 0 ⇐⇒ jξi + iξj = 0, ξ: skew-symmetric ⇐⇒ g( X ξ, Y) + g( Y ξ, X) = 0 ⇐⇒ g((φA − Aφ)X, Y) = 0 for any X, Y∈Γ(M). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow In the future homeogeneous hypersurfaces in HSSP satisfying certain geometric conditions might be solved completely as follows: Problem 1 Classify all of homogeneous hypersurfaces in HSSP. In this talk let us consider hypersurfaces with isometric Reeb flow in Hermitian Symmetric Spaces as follows: Problem 2 If M is a complete hypersurface in HSSP ¯M with isometric Reeb flow, then M becomes homogeneous ? Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow In the future homeogeneous hypersurfaces in HSSP satisfying certain geometric conditions might be solved completely as follows: Problem 1 Classify all of homogeneous hypersurfaces in HSSP. In this talk let us consider hypersurfaces with isometric Reeb flow in Hermitian Symmetric Spaces as follows: Problem 2 If M is a complete hypersurface in HSSP ¯M with isometric Reeb flow, then M becomes homogeneous ? Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Note 1) In CPm, CHm and QPm with isometric Reeb flow (See Okumura 1976, Montil and Romero 1986, Perez and Martinez 1986 ). Note 2) In G2(Cm+2), G∗ 2(Cm+2) and complex quadric Qm = SO(m + 2)/SO(2)SO(m) with isometric Reeb flow (See Berndt and Suh, 2002 and 2012, Suh, 2013, Berndt and Suh, 2013 ). Note 3) In near future, in noncompact complex quadric Qm∗ = SO(2, m)/SO(2)SO(m) with isometric Reeb flow will be classified. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Note 1) In CPm, CHm and QPm with isometric Reeb flow (See Okumura 1976, Montil and Romero 1986, Perez and Martinez 1986 ). Note 2) In G2(Cm+2), G∗ 2(Cm+2) and complex quadric Qm = SO(m + 2)/SO(2)SO(m) with isometric Reeb flow (See Berndt and Suh, 2002 and 2012, Suh, 2013, Berndt and Suh, 2013 ). Note 3) In near future, in noncompact complex quadric Qm∗ = SO(2, m)/SO(2)SO(m) with isometric Reeb flow will be classified. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Note 1) In CPm, CHm and QPm with isometric Reeb flow (See Okumura 1976, Montil and Romero 1986, Perez and Martinez 1986 ). Note 2) In G2(Cm+2), G∗ 2(Cm+2) and complex quadric Qm = SO(m + 2)/SO(2)SO(m) with isometric Reeb flow (See Berndt and Suh, 2002 and 2012, Suh, 2013, Berndt and Suh, 2013 ). Note 3) In near future, in noncompact complex quadric Qm∗ = SO(2, m)/SO(2)SO(m) with isometric Reeb flow will be classified. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Complex Projective Space Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Montiel and Romero classified hypersurfaces in CHm with isometric Reeb flow as follows: Theorem 1.1 (Montiel and Romero 1986) Let M be a real hypersurfaces in CHm with isometric Reeb flow. Then we have the following (A) M is an open part of a tube around a totally geodesic CHk in CHm, (C) geodesic hypersphere, (D) horosphere. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Montiel and Romero classified hypersurfaces in CHm with isometric Reeb flow as follows: Theorem 1.1 (Montiel and Romero 1986) Let M be a real hypersurfaces in CHm with isometric Reeb flow. Then we have the following (A) M is an open part of a tube around a totally geodesic CHk in CHm, (C) geodesic hypersphere, (D) horosphere. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Montiel and Romero classified hypersurfaces in CHm with isometric Reeb flow as follows: Theorem 1.1 (Montiel and Romero 1986) Let M be a real hypersurfaces in CHm with isometric Reeb flow. Then we have the following (A) M is an open part of a tube around a totally geodesic CHk in CHm, (C) geodesic hypersphere, (D) horosphere. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Montiel and Romero classified hypersurfaces in CHm with isometric Reeb flow as follows: Theorem 1.1 (Montiel and Romero 1986) Let M be a real hypersurfaces in CHm with isometric Reeb flow. Then we have the following (A) M is an open part of a tube around a totally geodesic CHk in CHm, (C) geodesic hypersphere, (D) horosphere. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Complex Two-Plane Grassmannians Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow When the maximal complex subbundle C (resp. quaternionic subbundle) of M in G2(Cm+2) is invariant, that is AC⊂C (resp. AQ⊂Q) , we say M is Hopf (resp. curvature adapted). Berndt and Suh (Monat, 1999) have classified real hypersurfaces in G2(Cm+2) as follows: Theorem 1.2 A real hypersurface of G2(Cm+2), m≥3, is Hopf and curvature adapted if and only if it is congruent to (A) a tube over a totally geodesic G2(Cm+1) in G2(Cm+2), (B) a tube over a totally geodesic totally real QPn, m = 2n, in G2(Cm+2). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow When the maximal complex subbundle C (resp. quaternionic subbundle) of M in G2(Cm+2) is invariant, that is AC⊂C (resp. AQ⊂Q) , we say M is Hopf (resp. curvature adapted). Berndt and Suh (Monat, 1999) have classified real hypersurfaces in G2(Cm+2) as follows: Theorem 1.2 A real hypersurface of G2(Cm+2), m≥3, is Hopf and curvature adapted if and only if it is congruent to (A) a tube over a totally geodesic G2(Cm+1) in G2(Cm+2), (B) a tube over a totally geodesic totally real QPn, m = 2n, in G2(Cm+2). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow When the maximal complex subbundle C (resp. quaternionic subbundle) of M in G2(Cm+2) is invariant, that is AC⊂C (resp. AQ⊂Q) , we say M is Hopf (resp. curvature adapted). Berndt and Suh (Monat, 1999) have classified real hypersurfaces in G2(Cm+2) as follows: Theorem 1.2 A real hypersurface of G2(Cm+2), m≥3, is Hopf and curvature adapted if and only if it is congruent to (A) a tube over a totally geodesic G2(Cm+1) in G2(Cm+2), (B) a tube over a totally geodesic totally real QPn, m = 2n, in G2(Cm+2). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Berndt and Suh (Monat. 2002) have given a classification of hypersurfaces in G2(Cm+2), m≥3 wih isometric Reeb flow as follows: Theorem 1.3 Let M be a real hypersurface in G2(Cm+2), m≥3, with isometric Reeb flow. Then M is locally congruent to (A) a tube over a totally geodesic G2(Cm+1) in G2(Cm+2). The two singular orbits are totally geodesically embedded CPm and G2(Cm+1), Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Homogeneous Hypersurfaces Isometric Reeb Flow Berndt and Suh (Monat. 2002) have given a classification of hypersurfaces in G2(Cm+2), m≥3 wih isometric Reeb flow as follows: Theorem 1.3 Let M be a real hypersurface in G2(Cm+2), m≥3, with isometric Reeb flow. Then M is locally congruent to (A) a tube over a totally geodesic G2(Cm+1) in G2(Cm+2). The two singular orbits are totally geodesically embedded CPm and G2(Cm+1), Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow The Riemannian symmetric space SU(2, m)/S(U(2)×U(m)) is a connected, simply connected, irreducible Riemannian symmetric space of noncompact type with rank 2. Let G = SU(2, m) and K = S(U(2)×U(m)), and denote by G and K the corresponding Lie algebra. Let B denotes the Cartan Killing form of G and by P the orthogonal complement of K in G with respect to B. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow The decomposition G = K⊕P is a Cartan decomposition of G = su(2, m). The Cartan involution θ∈Aut(g) on su(2, m) is given by θ(A) = I2,mAI2,m for A∈su(2, m), where I2,m = −I2 02,m 0m,2 Im Then < X, Y >= −B(X, θY): a positive definite Ad(K)-invariant on G. Its restriction to P: a Riemannian metrc g, where g: the Killing metric on SU(2, m)/S(U(2)×U(m)). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Killing Cartan forms related to sl(n, C) The Killing Cartan form B(X, Y) of sl(n, C) is given by B(X, Y) = 2nTrXY for any X, Y∈sl(n, C). In su(m + 2) = {X∈M(m + 2, C)|X∗ + X = 0, TrX = 0}, B(X, Y) is negative definite, because B(X, X) = −2nTrXX∗≤0. So < X, Y >= −B(X, Y). In su(2, m) = {X∈M(m + 2, C)|X∗I2,m + I2,mX = 0, TrX = 0}, the product < X, Y >= −B(X, θY), θ2 = I, is positive definite, because < X, X > = −2nTrXθX = −2nTrXI2,mXI2,m = 2nTrXX∗ I2 2,m = 2nTrXX∗ ≥0. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Killing Cartan forms related to sl(n, C) The Killing Cartan form B(X, Y) of sl(n, C) is given by B(X, Y) = 2nTrXY for any X, Y∈sl(n, C). In su(m + 2) = {X∈M(m + 2, C)|X∗ + X = 0, TrX = 0}, B(X, Y) is negative definite, because B(X, X) = −2nTrXX∗≤0. So < X, Y >= −B(X, Y). In su(2, m) = {X∈M(m + 2, C)|X∗I2,m + I2,mX = 0, TrX = 0}, the product < X, Y >= −B(X, θY), θ2 = I, is positive definite, because < X, X > = −2nTrXθX = −2nTrXI2,mXI2,m = 2nTrXX∗ I2 2,m = 2nTrXX∗ ≥0. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Killing Cartan forms related to sl(n, C) The Killing Cartan form B(X, Y) of sl(n, C) is given by B(X, Y) = 2nTrXY for any X, Y∈sl(n, C). In su(m + 2) = {X∈M(m + 2, C)|X∗ + X = 0, TrX = 0}, B(X, Y) is negative definite, because B(X, X) = −2nTrXX∗≤0. So < X, Y >= −B(X, Y). In su(2, m) = {X∈M(m + 2, C)|X∗I2,m + I2,mX = 0, TrX = 0}, the product < X, Y >= −B(X, θY), θ2 = I, is positive definite, because < X, X > = −2nTrXθX = −2nTrXI2,mXI2,m = 2nTrXX∗ I2 2,m = 2nTrXX∗ ≥0. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Let C = {X∈TM|JX∈TM} : the maximal complex subbundle and Q = {X∈TM|JX⊂TM} the maximal quaternionic subbundle for M in SU(2, m)/S(U(2)×U(m)). When C and Q of TM are both invariant by the shape operator A of M , we write h(C, C⊥ ) = 0 and h(Q, Q⊥ ) = 0, where h denotes the second fundamental form defined by g(h(X, Y), N) = g(AX, Y) for any X, Y on M. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow By using the theory of Focal points and the method due to P.B. Eberlein, Berndt and Suh proved the following (See Int. J. Math., 2012) Theorem 2.1 Let M be a connected hypersurface in SU2,m/S(U2Um), m≥2. Then h(C, C⊥) = 0 and h(Q, Q⊥) = 0 if and only if M is congruent to an open part of the following: (A) a tube around a totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) , or (B) a tube around a totally geodesic HHn in SU2,m/S(U2Um), m = 2n, (C) a horosphere whose center at infinity is singular . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow By using the theory of Focal points and the method due to P.B. Eberlein, Berndt and Suh proved the following (See Int. J. Math., 2012) Theorem 2.1 Let M be a connected hypersurface in SU2,m/S(U2Um), m≥2. Then h(C, C⊥) = 0 and h(Q, Q⊥) = 0 if and only if M is congruent to an open part of the following: (A) a tube around a totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) , or (B) a tube around a totally geodesic HHn in SU2,m/S(U2Um), m = 2n, (C) a horosphere whose center at infinity is singular . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow By using the theory of Focal points and the method due to P.B. Eberlein, Berndt and Suh proved the following (See Int. J. Math., 2012) Theorem 2.1 Let M be a connected hypersurface in SU2,m/S(U2Um), m≥2. Then h(C, C⊥) = 0 and h(Q, Q⊥) = 0 if and only if M is congruent to an open part of the following: (A) a tube around a totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) , or (B) a tube around a totally geodesic HHn in SU2,m/S(U2Um), m = 2n, (C) a horosphere whose center at infinity is singular . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow By using the theory of Focal points and the method due to P.B. Eberlein, Berndt and Suh proved the following (See Int. J. Math., 2012) Theorem 2.1 Let M be a connected hypersurface in SU2,m/S(U2Um), m≥2. Then h(C, C⊥) = 0 and h(Q, Q⊥) = 0 if and only if M is congruent to an open part of the following: (A) a tube around a totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) , or (B) a tube around a totally geodesic HHn in SU2,m/S(U2Um), m = 2n, (C) a horosphere whose center at infinity is singular . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Horosphere Let Ht = cos te1 + sin te2∈A: a unit normal to a horosphere Mt , where A denotes a maximal abelian subspace of P for the E. Cartan’s decomposition G = K⊕P. Here a horosphere is given by Mt = SHt ·o, where SHt denotes the Lie subgroup of G corresponding to the Lie subalgebra SH = S RH, S = A⊕N and N = ⊕λ∈Σ+ Gλ for the Iwasawa decomposition G = K⊕A⊕N with corresponding G = KAN. The shape operator of a horosphere Mt is given by AH = ad(H). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Characterization of type (A) and a Horosphere In this subsection we introduce a classification with isometric Reeb flow in SU2,m/S(U2Um) as follows (See Suh, Advances in Applied Math., 2013): Theorem 2.5 Let M be a connected orientable real hypersurface in SU2,m/S(U2Um), m ≥ 3. Then the Reeb flow on M is isometric if and only if M is congruent to an open part of the following: (A) a tube around some totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) or, (C) a horosphere whose center at infinity is singular. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Characterization of type (A) and a Horosphere In this subsection we introduce a classification with isometric Reeb flow in SU2,m/S(U2Um) as follows (See Suh, Advances in Applied Math., 2013): Theorem 2.5 Let M be a connected orientable real hypersurface in SU2,m/S(U2Um), m ≥ 3. Then the Reeb flow on M is isometric if and only if M is congruent to an open part of the following: (A) a tube around some totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) or, (C) a horosphere whose center at infinity is singular. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Characterization of type (A) and a Horosphere In this subsection we introduce a classification with isometric Reeb flow in SU2,m/S(U2Um) as follows (See Suh, Advances in Applied Math., 2013): Theorem 2.5 Let M be a connected orientable real hypersurface in SU2,m/S(U2Um), m ≥ 3. Then the Reeb flow on M is isometric if and only if M is congruent to an open part of the following: (A) a tube around some totally geodesic SU2,m−1/S(U2Um−1) in SU2,m/S(U2Um) or, (C) a horosphere whose center at infinity is singular. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Characterization of type (B) and a Horosphere Definition For a real hypersurface M in SU2,m/S(U2Um) is said to be a contact ⇐⇒ ∃ a non-zero constant function ρ defined on M such that φA + Aφ = kφ, k = 2ρ, The condition is equivalent to g((φA + Aφ)X, Y) = 2dη(X, Y), where dη is defined by dη(X, Y) = ( X η)Y − ( Y η)X for any X, Y on M in SU2,m/S(U2Um). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Then we give another classification in noncompact complex two-plane Grassmannian SU2,m/S(U2Um) in terms of the contact hypersurface as follows: Theorem 2.6 Let M be a contact real hypersurface in SU2,m/S(U2Um) with constant mean curvature. Then one of the following statements holds: (B) M is an open part of a tube around a totally geodesic HHn in SU2,2n/S(U2U2n), m = 2n, (C) M is an open part of a horosphere in SU2,m/S(U2Um) whose center at infinity is singular and of type JN ⊥ JN. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Then we give another classification in noncompact complex two-plane Grassmannian SU2,m/S(U2Um) in terms of the contact hypersurface as follows: Theorem 2.6 Let M be a contact real hypersurface in SU2,m/S(U2Um) with constant mean curvature. Then one of the following statements holds: (B) M is an open part of a tube around a totally geodesic HHn in SU2,2n/S(U2U2n), m = 2n, (C) M is an open part of a horosphere in SU2,m/S(U2Um) whose center at infinity is singular and of type JN ⊥ JN. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow Then we give another classification in noncompact complex two-plane Grassmannian SU2,m/S(U2Um) in terms of the contact hypersurface as follows: Theorem 2.6 Let M be a contact real hypersurface in SU2,m/S(U2Um) with constant mean curvature. Then one of the following statements holds: (B) M is an open part of a tube around a totally geodesic HHn in SU2,2n/S(U2U2n), m = 2n, (C) M is an open part of a horosphere in SU2,m/S(U2Um) whose center at infinity is singular and of type JN ⊥ JN. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem The Reeb flow on a real hypersurface in G2(Cm+2) is isometric if and only if M is an open part of a tube around a totally geodesic G2(Cm+1) ⊂ G2(Cm+2). In view of the previous results a natural expectation would lead to the totally geodesic Qm−1 ⊂ Qm. Surprisingly, this is not the case. In fact, we will prove Theorem 3.1 Let M be a real hypersurface of the complex quadric Qm, m ≥ 3. The Reeb flow on M is isometric if and only if m is even, say m = 2k, and M is an open part of a tube around a totally geodesic CPk ⊂ Q2k . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem The homogeneous quadratic equation Qm = {z∈Cm+2 |z2 1 + . . . + z2 m+2 = 0}⊂CPm+1 defines a complex hypersurface in complex projective space CPm+1 = SUm+2/S(Um+1U1). For a unit normal vector N of Qm at a point [z] ∈ Qm we denote by AN the shape operator of Qm in CPm+1 with respect to N. The shape operator is an involution on T[z]Qm and T[z]Qm = V(AN) ⊕ JV(AN), where V(AN) is the (+1)-eigenspace and JV(AN) is the (−1)-eigenspace of AN. Geometrically this means that AN defines a real structure on the complex vector space T[z]Qm, or equivalently, is a complex conjugation on T[z]Qm. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem The Riemannian curvature tensor ¯R of Qm can be expressed as follows: ¯R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX −g(JX, Z)JY − 2g(JX, Y)JZ +g(AY, Z)AX − g(AX, Z)AY +g(JAY, Z)JAX − g(JAX, Z)JAY. A nonzero tangent vector W ∈ T[z]Qm is called singular if it is tangent to more than one maximal flat in Qm. 1. If a conjugation A ∈ A[z] such that W ∈ V(A), then W is singular, that is A-principal. 2. If a conjugation A ∈ A[z] and orthonormal vectors X, Y ∈ V(A) such that W/||W|| = (X + JY)/ √ 2, then W is said to be A-isotropic. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem The Riemannian curvature tensor ¯R of Qm can be expressed as follows: ¯R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX −g(JX, Z)JY − 2g(JX, Y)JZ +g(AY, Z)AX − g(AX, Z)AY +g(JAY, Z)JAX − g(JAX, Z)JAY. A nonzero tangent vector W ∈ T[z]Qm is called singular if it is tangent to more than one maximal flat in Qm. 1. If a conjugation A ∈ A[z] such that W ∈ V(A), then W is singular, that is A-principal. 2. If a conjugation A ∈ A[z] and orthonormal vectors X, Y ∈ V(A) such that W/||W|| = (X + JY)/ √ 2, then W is said to be A-isotropic. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let M be a real hypersurface of Qm and denote ξ = −JN, where N is a (local) unit normal vector field of M. For A ∈ A[z] and X ∈ T[z]M we decompose AX as follows: AX = BX + ρ(X)N where BX is the tangential component of AX and ρ(X) = g(AX, N) = g(X, AN) = g(X, AJξ) = −g(X, JAξ) = g(JX, Aξ). Since JX = φX + η(X)N and Aξ = Bξ + ρ(ξ)N we also have ρ(X) = g(φX, Bξ) + η(X)ρ(ξ) = g(−φBξ + ρ(ξ)ξ, X). We also define δ = g(N, AN) = g(JN, JAN) = −g(JN, AJN) = −g(ξ, Aξ). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let M be a real hypersurface of Qm and denote ξ = −JN, where N is a (local) unit normal vector field of M. For A ∈ A[z] and X ∈ T[z]M we decompose AX as follows: AX = BX + ρ(X)N where BX is the tangential component of AX and ρ(X) = g(AX, N) = g(X, AN) = g(X, AJξ) = −g(X, JAξ) = g(JX, Aξ). Since JX = φX + η(X)N and Aξ = Bξ + ρ(ξ)N we also have ρ(X) = g(φX, Bξ) + η(X)ρ(ξ) = g(−φBξ + ρ(ξ)ξ, X). We also define δ = g(N, AN) = g(JN, JAN) = −g(JN, AJN) = −g(ξ, Aξ). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let M be a real hypersurface of Qm and denote ξ = −JN, where N is a (local) unit normal vector field of M. For A ∈ A[z] and X ∈ T[z]M we decompose AX as follows: AX = BX + ρ(X)N where BX is the tangential component of AX and ρ(X) = g(AX, N) = g(X, AN) = g(X, AJξ) = −g(X, JAξ) = g(JX, Aξ). Since JX = φX + η(X)N and Aξ = Bξ + ρ(ξ)N we also have ρ(X) = g(φX, Bξ) + η(X)ρ(ξ) = g(−φBξ + ρ(ξ)ξ, X). We also define δ = g(N, AN) = g(JN, JAN) = −g(JN, AJN) = −g(ξ, Aξ). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Geometric Descriptions of the Tube We assume that m is even, say m = 2k. The map CPk → Q2k ⊂ CP2k+1 , [z1, . . . , zk+1] → [z1, . . . , zk+1, iz1, . . . , izk+1] gives an embedding of CPk into Q2k as a totally geodesic complex submanifold. Define a complex structure j on C2k+2 by j(z1, . . . , zk+1, zk+2, . . . , z2k+2) = (−zk+2, . . . , −z2k+2, z1, . . . , zk+1). Then j2 = −I and note that ij = ji. We can then identify C2k+2 with Ck+1 ⊕ jCk+1 and get T[z]CPk = {X + jiX | X ∈ Ck+1 [z]} = {X + ijX|X∈V(A¯z)}. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem The normal space becomes ν[z]CPk = A¯z(T[z]CPk ) = {X − ijX|X∈V(A¯z)}. The normal N of T[z]CPk : A-isotropic, the four vectors {N, JN, AN, JAN}: pairwise orthonormal. The normal Jacobi operator ¯RN is given by ¯RNZ = ¯R(Z, N)N = Z − g(Z, N)N + 3g(Z, JN)JN −g(Z, AN)AN − g(Z, JAN)JAN. Both T[z]CPk and ν[z]CPk are invariant under RN, and RN has three eigenvalues 0, 1, 4 according to RN⊕[AN], T[z]Q2k ([N]⊕[AN]) and RJN. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Principal Curvatures and Spaces of the Tube To calculate the principal curvatures of the tube of radius 0 < r < π/2 around CPk : the standard Jacobi field method as described in Section 8.2 of Berndt, Console and Olmos. Let γ: the geodesic in Q2k with γ(0) = [z] and ˙γ(0) = N. γ⊥ : the parallel subbundle of TQ2k along γ defined by γ⊥ γ(t) = T[γ(t)]Q2k R˙γ(t). Let us define the γ⊥-valued tensor field R⊥ γ along γ by R⊥ γ(t)X = R(X, ˙γ(t))˙γ(t). Now consider the End(γ⊥)-valued differential equation Y + R⊥ γ ◦ Y = 0. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let D be the unique solution of this differential equation with initial values D(0) = I 0 0 0 , D (0) = 0 0 0 I , where the decomposition of the matrices is with respect to γ⊥ [z] = T[z]CPk ⊕ (ν[z]CPk RN) and I denotes the identity transformation on the corresponding space. Then the shape operator S(r) of the tube of radius 0 < r < π/2 around CPk with respect to ˙γ(r) is given by S(r) = −D (r) ◦ D−1 (r). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let D be the unique solution of this differential equation with initial values D(0) = I 0 0 0 , D (0) = 0 0 0 I , where the decomposition of the matrices is with respect to γ⊥ [z] = T[z]CPk ⊕ (ν[z]CPk RN) and I denotes the identity transformation on the corresponding space. Then the shape operator S(r) of the tube of radius 0 < r < π/2 around CPk with respect to ˙γ(r) is given by S(r) = −D (r) ◦ D−1 (r). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Let D be the unique solution of this differential equation with initial values D(0) = I 0 0 0 , D (0) = 0 0 0 I , where the decomposition of the matrices is with respect to γ⊥ [z] = T[z]CPk ⊕ (ν[z]CPk RN) and I denotes the identity transformation on the corresponding space. Then the shape operator S(r) of the tube of radius 0 < r < π/2 around CPk with respect to ˙γ(r) is given by S(r) = −D (r) ◦ D−1 (r). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem If we decompose γ⊥ [z] further into γ⊥ [z] = (T[z]CPk [AN]) ⊕ [AN] ⊕ (ν[z]CPk [N]) ⊕ RJN, we get by explicit computation that S(r) =     0 0 0 0 0 tan(r) 0 0 0 0 − cot(r) 0 0 0 0 −2 cot(2r)     with respect to that decomposition. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem If we decompose γ⊥ [z] further into γ⊥ [z] = (T[z]CPk [AN]) ⊕ [AN] ⊕ (ν[z]CPk [N]) ⊕ RJN, we get by explicit computation that S(r) =     0 0 0 0 0 tan(r) 0 0 0 0 − cot(r) 0 0 0 0 −2 cot(2r)     with respect to that decomposition. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Proposition 3.1 Let M be the tube of radius 0 < r < π/2 around the totally geodesic CPk in Q2k . Then the following hold: 1. M is a Hopf hypersurface. 2. The normal bundle of M consists of A-isotropic singular. 3. M has four distinct constant principal curvatures. principal curvature eigenspace multiplicity 0 C Q 2 tan(r) TCPk (C Q) 2k − 2 − cot(r) νCPk CνM 2k − 2 −2 cot(2r) F 1 4. Sφ = φS. 5. The Reeb flow on M is an isometric flow. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem 1 Introduction Homogeneous Hypersurfaces Isometric Reeb Flow 2 Hyperbolic Grassmannians Hypersurfaces in SU2,m/S(U2Um) Isometric Reeb Flow 3 Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Now we investigate real hypersurfaces in Qm for which the Reeb flow is isometric. From this, we get a complete expression for the covariant derivative as follows: ( X S)Y = {dα(X)η(Y) + g((αSφ − S2 φ)X, Y) +δη(Y)ρ(X) + δg(BX, φY) + η(BX)ρ(Y)}ξ +{η(Y)ρ(X) + g(BX, φY)}Bξ + g(BX, Y)φBξ −ρ(Y)BX − η(Y)φX − η(BY)φBX. Lemma 3.1 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then the normal vector field N is A-isotropic everywhere. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem From Proposition and Lemma the principal curvature function α is constant. Then we get (λ2 − αλ)Y + (λ2 − αλ)Z = (S2 − αS)X = Y. By virtue of this equation, we can assert the following propositions: Proposition 3.2 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then the distributions Q and C Q = [Bξ] are invariant. Proposition 3.3 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then m is even, say m = 2k, and the real structure A maps Tλ onto Tµ, and vice versa. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem From Proposition and Lemma the principal curvature function α is constant. Then we get (λ2 − αλ)Y + (λ2 − αλ)Z = (S2 − αS)X = Y. By virtue of this equation, we can assert the following propositions: Proposition 3.2 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then the distributions Q and C Q = [Bξ] are invariant. Proposition 3.3 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then m is even, say m = 2k, and the real structure A maps Tλ onto Tµ, and vice versa. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem From Proposition and Lemma the principal curvature function α is constant. Then we get (λ2 − αλ)Y + (λ2 − αλ)Z = (S2 − αS)X = Y. By virtue of this equation, we can assert the following propositions: Proposition 3.2 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then the distributions Q and C Q = [Bξ] are invariant. Proposition 3.3 Let M be a real hypersurface in Qm, m ≥ 3, with isometric Reeb flow. Then m is even, say m = 2k, and the real structure A maps Tλ onto Tµ, and vice versa. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem For each point [z] ∈ M we denote by γ[z] the geodesic in Q2k with γ[z](0) = [z] and ˙γ[z](0) = N[z] and by F the smooth map F : M −→ Qm , [z] −→ γ[z](r). F is the displacement of M at distance r in the direction of N. Thee differential d[z]F of F at [z] can be computed by d[z]F(X) = ZX (r), where ZX is the Jacobi vector field along γ[z] with ZX (0) = X and ZX (0) = −SX. The A-isotropic N gives that RN = R(Z, N)N has the three constant eigenvalues 0, 1, 4 with corresponding eigenbundles νM ⊕ (C Q) = νM ⊕ Tν, Q = Tλ ⊕ Tµ and F = Tα. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem Rigidity of totally geodesic submanifolds : =⇒ M is an open part of a tube of radius r around a k-dimensional connected, complete, totally geodesic complex submanifold P of Q2k . Klein classified the totally geodesic submanifolds P in Q2k as follows: The focal submanifold P : a totally geodesic Qk ⊂ Q2k or a totally geodesic CPk ⊂ Q2k . ⇐⇒ M is an open part of a tube around CPk . Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 127(1999), 1-14. J. Berndt and Y.J. Suh, Isometric flows on real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 137(2002), 87-98. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(2006), 355-364. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 127(1999), 1-14. J. Berndt and Y.J. Suh, Isometric flows on real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 137(2002), 87-98. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(2006), 355-364. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 127(1999), 1-14. J. Berndt and Y.J. Suh, Isometric flows on real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 137(2002), 87-98. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(2006), 355-364. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 127(1999), 1-14. J. Berndt and Y.J. Suh, Isometric flows on real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 137(2002), 87-98. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(2006), 355-364. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References II J.D. Perez and Y.J. Suh, Real hypersurfaces of quaternionic projective space satisfying Ui R = 0, Diff. Geom. Appl. 7(1997), 211-217. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor, J. of Geom. and Physics, 60(2010), 1792-1805. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Royal Soc. Edinburgh 142(A)(2012), 1309-1324. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References II J.D. Perez and Y.J. Suh, Real hypersurfaces of quaternionic projective space satisfying Ui R = 0, Diff. Geom. Appl. 7(1997), 211-217. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor, J. of Geom. and Physics, 60(2010), 1792-1805. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Royal Soc. Edinburgh 142(A)(2012), 1309-1324. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References II J.D. Perez and Y.J. Suh, Real hypersurfaces of quaternionic projective space satisfying Ui R = 0, Diff. Geom. Appl. 7(1997), 211-217. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor, J. of Geom. and Physics, 60(2010), 1792-1805. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Royal Soc. Edinburgh 142(A)(2012), 1309-1324. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References III Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor, J. of Geom. and Physics, 64(2013), 1-11. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, Journal de Math. Pures Appl., 100(2013), 16-33. J. Berndt and Y.J. Suh, Real hypersurfaces in the noncompact Grassmannians SU2,m/S(U2·Um), http://arxiv.org/abs/0911.3081, International J. of Math., World Sci. Publ., 23(2012), 1250103(35 pages). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References III Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor, J. of Geom. and Physics, 64(2013), 1-11. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, Journal de Math. Pures Appl., 100(2013), 16-33. J. Berndt and Y.J. Suh, Real hypersurfaces in the noncompact Grassmannians SU2,m/S(U2·Um), http://arxiv.org/abs/0911.3081, International J. of Math., World Sci. Publ., 23(2012), 1250103(35 pages). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References III Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor, J. of Geom. and Physics, 64(2013), 1-11. Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, Journal de Math. Pures Appl., 100(2013), 16-33. J. Berndt and Y.J. Suh, Real hypersurfaces in the noncompact Grassmannians SU2,m/S(U2·Um), http://arxiv.org/abs/0911.3081, International J. of Math., World Sci. Publ., 23(2012), 1250103(35 pages). Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References IV J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatschefte fur Math. 137(2002), 87-98. Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Advances in Applied Mathematics, 50(2013), 645-659. J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, International J. Math., 24(2013),(in press). J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Research Notes in Mathematics 434, Chapman & Hall/CRC, 2003. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References IV J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatschefte fur Math. 137(2002), 87-98. Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Advances in Applied Mathematics, 50(2013), 645-659. J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, International J. Math., 24(2013),(in press). J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Research Notes in Mathematics 434, Chapman & Hall/CRC, 2003. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References IV J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatschefte fur Math. 137(2002), 87-98. Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Advances in Applied Mathematics, 50(2013), 645-659. J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, International J. Math., 24(2013),(in press). J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Research Notes in Mathematics 434, Chapman & Hall/CRC, 2003. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References IV J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatschefte fur Math. 137(2002), 87-98. Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Advances in Applied Mathematics, 50(2013), 645-659. J. Berndt and Y.J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, International J. Math., 24(2013),(in press). J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Research Notes in Mathematics 434, Chapman & Hall/CRC, 2003. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References V P.B. Eberlein, Geometry of Non positively Curved Manifolds, Chicago Lectures in Math., The Univ. of Chicago Press, 1996, A.W. Knapp, Lie Groups beyond an Introduction, Progress in Math., Birkhäuser, 2002, S. Helgason, Differential Geometry, Lie Group and Symmetric Spaces, Graduate Studies in Mathematics 34, Amer. Math. Soc. 2001, S. Helgason, Geometric Analysis on Symmetric Spaces, The 2nd Edition, Math. Survey and Monographs 39, Amer. Math. Soc. 2008. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References V P.B. Eberlein, Geometry of Non positively Curved Manifolds, Chicago Lectures in Math., The Univ. of Chicago Press, 1996, A.W. Knapp, Lie Groups beyond an Introduction, Progress in Math., Birkhäuser, 2002, S. Helgason, Differential Geometry, Lie Group and Symmetric Spaces, Graduate Studies in Mathematics 34, Amer. Math. Soc. 2001, S. Helgason, Geometric Analysis on Symmetric Spaces, The 2nd Edition, Math. Survey and Monographs 39, Amer. Math. Soc. 2008. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References V P.B. Eberlein, Geometry of Non positively Curved Manifolds, Chicago Lectures in Math., The Univ. of Chicago Press, 1996, A.W. Knapp, Lie Groups beyond an Introduction, Progress in Math., Birkhäuser, 2002, S. Helgason, Differential Geometry, Lie Group and Symmetric Spaces, Graduate Studies in Mathematics 34, Amer. Math. Soc. 2001, S. Helgason, Geometric Analysis on Symmetric Spaces, The 2nd Edition, Math. Survey and Monographs 39, Amer. Math. Soc. 2008. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem References V P.B. Eberlein, Geometry of Non positively Curved Manifolds, Chicago Lectures in Math., The Univ. of Chicago Press, 1996, A.W. Knapp, Lie Groups beyond an Introduction, Progress in Math., Birkhäuser, 2002, S. Helgason, Differential Geometry, Lie Group and Symmetric Spaces, Graduate Studies in Mathematics 34, Amer. Math. Soc. 2001, S. Helgason, Geometric Analysis on Symmetric Spaces, The 2nd Edition, Math. Survey and Monographs 39, Amer. Math. Soc. 2008. Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces Introduction Hyperbolic Grassmannians Complex Quadrics Real hypersurfaces in Q2k Tubes around the totally geodesic CPk ⊂ Q2k Proof of Main Theorem ENDING THANKS FOR YOUR ATTENTION! Y.J.Suh Isometric Reeb Flow on Hermitian Symmetric Spaces