Geometric constructions related to isoparametric foliations Chao - - PowerPoint PPT Presentation
Geometric constructions related to isoparametric foliations Chao Qian Beijing Institute of Technology Based on the joint works with Prof. Peng and Tang 02. 06. 2019 Chao Qian Geometric constructions related to isoparametric foliations
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves
1
Introduction Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
2
Related constructions A generalization by Q.-Tang Pinkall-Thorbergsson Construction
3
Curvature of leaves Ricci curvature sectional curvature
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
1
Introduction Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
2
Related constructions A generalization by Q.-Tang Pinkall-Thorbergsson Construction
3
Curvature of leaves Ricci curvature sectional curvature
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Let N be a connected complete Riemannian manifold. Definition A transnormal system F is a subdivision of N into submanifolds, the so-called leaves, such that geodesics perpendicular to one leaf stay perpendicular to all. Moreover, if there are vertical vector fields {Xi ∈ Γ(TN)|i ∈ I} such that TpL(p) = span{Xi|p|i ∈ I}, (1) for any p ∈ N and L(p) the leaf through p, then (N, F) is said to be a singular Riemannian foliation.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Let N be a connected complete Riemannian manifold. Definition A transnormal system F is a subdivision of N into submanifolds, the so-called leaves, such that geodesics perpendicular to one leaf stay perpendicular to all. Moreover, if there are vertical vector fields {Xi ∈ Γ(TN)|i ∈ I} such that TpL(p) = span{Xi|p|i ∈ I}, (1) for any p ∈ N and L(p) the leaf through p, then (N, F) is said to be a singular Riemannian foliation.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Definition Let N be a closed Riemannian manifold. A smooth function f : N → R is said to be transnormal , if there is a smooth function b such that |∇f|2 = b(f). (2) If moreover ∆f = a(f) (3) for a function a, then f is called isoparametric. Write Imf = [α, β]. According to Wang[Math. Ann., 1987], (Imf)◦ has only regular value, and each level set f −1(t) is a smooth submanifold.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Definition Let N be a closed Riemannian manifold. A smooth function f : N → R is said to be transnormal , if there is a smooth function b such that |∇f|2 = b(f). (2) If moreover ∆f = a(f) (3) for a function a, then f is called isoparametric. Write Imf = [α, β]. According to Wang[Math. Ann., 1987], (Imf)◦ has only regular value, and each level set f −1(t) is a smooth submanifold.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Define M+ = f −1(β) and M− = f −1(α), the focal submanifolds. Actually, all the level sets form a singular Riemannian foliation of codimension 1, such that each regular leaf has CMC, the so-called isoparametric foliation (of codimension 1).
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Define M+ = f −1(β) and M− = f −1(α), the focal submanifolds. Actually, all the level sets form a singular Riemannian foliation of codimension 1, such that each regular leaf has CMC, the so-called isoparametric foliation (of codimension 1).
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Cartan: 1938: Isoparametric hypersurface in Rn+1, Sn+1 or Hn+1 ⇐ ⇒ hypersurface with CPC. 1939-1940: g = 3, tubes over Veronese embeddings of FP2 in S4, S7, S13, S25 for F = R, C, H, Ca.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
M¨ unzner: Mn, an isoparametric hypersurface in Sn+1 with constant principal curvatures cot θi, 0 < θ1 < · · · < θg < π, with multiplicities mi, then: 1). θk = θ1 + k−1
g π;
2). mk = mk+2(subscripts mod g); 3). M: open subset of level hypersurface in Sn+1 of a homogeneous polynomial F of degree g on Rn+2 satisfying Cartan-M¨ unzner equations: |∇F|2 = g2|x|2g−2, △F = m2−m1
2
g2|x|g−2; 4). g = 1, 2, 3, 4, 6.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Cecil-Chi-Jensen,S. Immervoll,Q. Chi: For g=4, isoparametric hypersurfaces must be OT-FKM type or homogeneous with (2,2), (4,5). Dorfmeister-Neher, g=6, m=1,
For g=6, isoparametric hypersurfaces are homogenous.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Cecil-Chi-Jensen,S. Immervoll,Q. Chi: For g=4, isoparametric hypersurfaces must be OT-FKM type or homogeneous with (2,2), (4,5). Dorfmeister-Neher, g=6, m=1,
For g=6, isoparametric hypersurfaces are homogenous.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
For a symmetric Clifford system {P0, · · · , Pm} on R2l, i.e., Pi’s are symmetric matrices satisfying PiPj + PjPi = 2δijI2l, Ferus-Karcher-M¨ unzner [Math. Z., 1981] constructed a polynomial F on R2l defined by F(x) = |x|4 − 2
m
Pix, x2. Then f = F|S2l−1 is an isoparametric function,i.e.,
△f = 8(m2 − m1) − 4(2l + 2)f. Consequently, each level hypersurface has g = 4 and (m1, m2) = (m, l − m − 1), the so-called OT-FKM type. Denote focal submanifolds by M+ = f −1(1) and M− = f −1(−1), which have codimension m1 + 1 and m2 + 1.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
For a symmetric Clifford system {P0, · · · , Pm} on R2l, i.e., Pi’s are symmetric matrices satisfying PiPj + PjPi = 2δijI2l, Ferus-Karcher-M¨ unzner [Math. Z., 1981] constructed a polynomial F on R2l defined by F(x) = |x|4 − 2
m
Pix, x2. Then f = F|S2l−1 is an isoparametric function,i.e.,
△f = 8(m2 − m1) − 4(2l + 2)f. Consequently, each level hypersurface has g = 4 and (m1, m2) = (m, l − m − 1), the so-called OT-FKM type. Denote focal submanifolds by M+ = f −1(1) and M− = f −1(−1), which have codimension m1 + 1 and m2 + 1.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Q.-Tang-Yan [A.G.A.G., 2013], Xie [Acta Math. Sinica, 2015]: M± are Willmore in unit spheres; Q.-Tang [P .L.M.S., 2016]: Focal maps from isoparametric hypersurfaces M to M± are harmonic; Q.-Tang [P .L.M.S., 2016]: For g = 4, M± are minimal with σ(M±) = max{ |B(X, X)|2 | X ∈ TM±, |X| = 1} = 1, which provide infinitely many counterexamples to Conjecture A and Conjecture B of P. F. Leung[Bull. L.M.S., 1991];
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Q.-Tang-Yan [A.G.A.G., 2013], Xie [Acta Math. Sinica, 2015]: M± are Willmore in unit spheres; Q.-Tang [P .L.M.S., 2016]: Focal maps from isoparametric hypersurfaces M to M± are harmonic; Q.-Tang [P .L.M.S., 2016]: For g = 4, M± are minimal with σ(M±) = max{ |B(X, X)|2 | X ∈ TM±, |X| = 1} = 1, which provide infinitely many counterexamples to Conjecture A and Conjecture B of P. F. Leung[Bull. L.M.S., 1991];
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Q.-Tang-Yan [A.G.A.G., 2013], Xie [Acta Math. Sinica, 2015]: M± are Willmore in unit spheres; Q.-Tang [P .L.M.S., 2016]: Focal maps from isoparametric hypersurfaces M to M± are harmonic; Q.-Tang [P .L.M.S., 2016]: For g = 4, M± are minimal with σ(M±) = max{ |B(X, X)|2 | X ∈ TM±, |X| = 1} = 1, which provide infinitely many counterexamples to Conjecture A and Conjecture B of P. F. Leung[Bull. L.M.S., 1991];
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
Q.-Tang-Yan [A.G.A.G., 2013], Xie [Acta Math. Sinica, 2015]: M± are Willmore in unit spheres; Q.-Tang [P .L.M.S., 2016]: Focal maps from isoparametric hypersurfaces M to M± are harmonic; Q.-Tang [P .L.M.S., 2016]: For g = 4, M± are minimal with σ(M±) = max{ |B(X, X)|2 | X ∈ TM±, |X| = 1} = 1, which provide infinitely many counterexamples to Conjecture A and Conjecture B of P. F. Leung[Bull. L.M.S., 1991];
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
1
Introduction Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
2
Related constructions A generalization by Q.-Tang Pinkall-Thorbergsson Construction
3
Curvature of leaves Ricci curvature sectional curvature
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Given a symmetric Clifford system {P0, · · · , Pm} on R2l with the Euclidean metric ·, ·, i.e. Pi’s are symmetric matrices satisfying PiPj + PjPi = 2δijI2l. Mi := {x ∈ S2l−1(1) | P0x, x = P1x, x = · · · = Pix, x = 0}, and Mm = M+ ⊂ Mm−1 ⊂ · · · ⊂ M0 ⊂ S2l−1(1). ∀0 ≤ i ≤ m − 1, define fi : Mi → R by fi(x) = Pi+1x, x for x ∈ Mi. Ni := {x ∈ S2l−1(1) | P0x, x2 + P1x, x2 + · · · + Pix, x2 = 1}, and N1 ⊂ N2 ⊂ · · · ⊂ Nm = M− ⊂ S2l−1(1). ∀2 ≤ i ≤ m, define gi : Ni → R by gi(x) = Pix, x for x ∈ Ni.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Given a symmetric Clifford system {P0, · · · , Pm} on R2l with the Euclidean metric ·, ·, i.e. Pi’s are symmetric matrices satisfying PiPj + PjPi = 2δijI2l. Mi := {x ∈ S2l−1(1) | P0x, x = P1x, x = · · · = Pix, x = 0}, and Mm = M+ ⊂ Mm−1 ⊂ · · · ⊂ M0 ⊂ S2l−1(1). ∀0 ≤ i ≤ m − 1, define fi : Mi → R by fi(x) = Pi+1x, x for x ∈ Mi. Ni := {x ∈ S2l−1(1) | P0x, x2 + P1x, x2 + · · · + Pix, x2 = 1}, and N1 ⊂ N2 ⊂ · · · ⊂ Nm = M− ⊂ S2l−1(1). ∀2 ≤ i ≤ m, define gi : Ni → R by gi(x) = Pix, x for x ∈ Ni.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Given a symmetric Clifford system {P0, · · · , Pm} on R2l with the Euclidean metric ·, ·, i.e. Pi’s are symmetric matrices satisfying PiPj + PjPi = 2δijI2l. Mi := {x ∈ S2l−1(1) | P0x, x = P1x, x = · · · = Pix, x = 0}, and Mm = M+ ⊂ Mm−1 ⊂ · · · ⊂ M0 ⊂ S2l−1(1). ∀0 ≤ i ≤ m − 1, define fi : Mi → R by fi(x) = Pi+1x, x for x ∈ Mi. Ni := {x ∈ S2l−1(1) | P0x, x2 + P1x, x2 + · · · + Pix, x2 = 1}, and N1 ⊂ N2 ⊂ · · · ⊂ Nm = M− ⊂ S2l−1(1). ∀2 ≤ i ≤ m, define gi : Ni → R by gi(x) = Pix, x for x ∈ Ni.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Theorem (Q. and Tang, P .L.M.S., 2016) (1). For 0 ≤ i ≤ m − 1, fi : Mi → R with Im(fi) = [−1, 1] is an isoparametric function satisfying |∇fi|2 = 4(1 − f 2
i ), △fi
= −4(l − i − 1)fi. ∀c ∈ (−1, 1), Uc = f −1
i
(c) has 3 principal curvatures. The focal submanifolds U±1 = f −1
i
(±1) are both isometric to Sl−1(1) and totally geodesic in Mi. (2). Similarly, for 2 ≤ i ≤ m, gi : Ni → R with Im(gi) = [−1, 1] is an isoparametric function satisfying |∇gi|2 = 4(1 − g2
i ), △gi
= −4igi. ∀c ∈ (−1, 1), Vc = g−1
i
(c) has 3 principal curvatures. The focal submanifolds V±1 = g−1
i
(±1) are both isometric to Sl−1(1) and totally geodesic in Ni.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Theorem (Q. and Tang, P .L.M.S., 2016) (1). For 0 ≤ i ≤ m − 1, fi : Mi → R with Im(fi) = [−1, 1] is an isoparametric function satisfying |∇fi|2 = 4(1 − f 2
i ), △fi
= −4(l − i − 1)fi. ∀c ∈ (−1, 1), Uc = f −1
i
(c) has 3 principal curvatures. The focal submanifolds U±1 = f −1
i
(±1) are both isometric to Sl−1(1) and totally geodesic in Mi. (2). Similarly, for 2 ≤ i ≤ m, gi : Ni → R with Im(gi) = [−1, 1] is an isoparametric function satisfying |∇gi|2 = 4(1 − g2
i ), △gi
= −4igi. ∀c ∈ (−1, 1), Vc = g−1
i
(c) has 3 principal curvatures. The focal submanifolds V±1 = g−1
i
(±1) are both isometric to Sl−1(1) and totally geodesic in Ni.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Remark 1). Mm ⊂ Mm−1 ⊂ · · · ⊂ M0 ⊂ S2l−1(1) is a minimal isoparametric sequence, i.e., each Mi+1 is a minimal isoparametric hypersurface in Mi for 0 ≤ i ≤ m − 1. Moreover, Mi+j is minimal in Mi. 2). Similarly, N1 ⊂ N2 ⊂ · · · ⊂ Nm ⊂ S2l−1(1) is also a minimal isoparametric sequence. Corollary (Q. and Tang, P .L.M.S., 2016) (1). For 0 ≤ i ≤ m − 1, each Mi+1 fibers over Sl−1 with fiber Sl−i−2. (2). For 2 ≤ i ≤ m, each Ni−1 fibers over Sl−1 with fiber Si−1.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Remark 1). Mm ⊂ Mm−1 ⊂ · · · ⊂ M0 ⊂ S2l−1(1) is a minimal isoparametric sequence, i.e., each Mi+1 is a minimal isoparametric hypersurface in Mi for 0 ≤ i ≤ m − 1. Moreover, Mi+j is minimal in Mi. 2). Similarly, N1 ⊂ N2 ⊂ · · · ⊂ Nm ⊂ S2l−1(1) is also a minimal isoparametric sequence. Corollary (Q. and Tang, P .L.M.S., 2016) (1). For 0 ≤ i ≤ m − 1, each Mi+1 fibers over Sl−1 with fiber Sl−i−2. (2). For 2 ≤ i ≤ m, each Ni−1 fibers over Sl−1 with fiber Si−1.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Let {E1, E2, ..., Em−1} be a set of orthogonal matrices on Rl with the Euclidean metric, which satisfy EαEβ + EβEα = −2δαβId for 1 ≤ α, β ≤ m − 1. Define P0 := Id −Id
Id Id
−Eα−1
for 2 ≤ α ≤ m. Then {P0, P1, ..., Pm} is a symmetric Clifford system on R2l. Pinkall-Thorbergsson[P .A.M.S., 1989]: for 0 < t ≤ π
4 , define
Mt
+ := {z = (x, y) ∈ Rl ⊕ Rl = R2l :
|x| = cos t, |y| = sin t, x, y = 0, x, Eαy = 0 for 1 ≤ α ≤ m − 1}.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Write a := tan t, b := cot t, Q0 := aId −bId
1 ≤ α ≤ m. Then Mt
+ = {z = (x, y) ∈ R2l | |x|2 + |y|2 = 1, z, Qαz = 0 for 0 ≤ α ≤ m}.
Theorem (Q. and Tang, arXiv, 2018) Given z ∈ Mt
+:
(1) The normal space NzMt
+ at z is given by
NzMt
+ = Span{Q0z, Q1z, · · · , Qmz}.
The tangent space TzMt
+ at z is given by
TzMt
+ = {(u, v) ∈ Rl ⊕ Rl | (u, v), z = 0, (u, v), Qiz = 0 for 0 ≤ i ≤ m}.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Theorem (Q. and Tang, arXiv, 2018) (2) For 1 ≤ α ≤ m, the shape operator Aα of Mt
+ with respect to Qαz
has principal curvatures 1, 0, −1 of multiplicities l − m − 1, m, and l − m − 1, respectively. Moreover, TzMt
+ = E+(Qαz) ⊕ E0(Qαz) ⊕ E−(Qαz),
where E+(Qαz) = E−(Qα) ∩ TzMt
+,
E0(Qαz) = Span{QαQβz | 0 ≤ β ≤ m, β = α}, and E−(Qαz) = E+(Qα) ∩ TzMt
+ are principal distributions of 1, 0, and −1,
respectively.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Theorem (Q. and Tang, arXiv, 2018) (3) Let E+(Q0) and E−(Q0) be eigenspaces of Q0 with eigenvalues a and −b, respectively. For the unit normal vector Q0z, the shape
l − m − 1, m, and l − m − 1, respectively. Moreover, TzMt
+ = E+(Q0z) ⊕ E0(Q0z) ⊕ E−(Q0z),
where E+(Q0z) = E−(Q0) ∩ TzMt
+,
E0(Q0z) = Span{Q−1
0 Qαz | 1 ≤ α ≤ m}, and E−(Q0z) = E+(Q0) ∩ TzMt +
are principal distributions of cot t, 0, and − tan t, respectively.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Remark 1). For any 0 < t < π
4 , both |H|2 and |B|2 of Mt + ⊂ S2l−1(1) are positive
constants. 2). The scalar curvature St of Mt
+ is a positive constant for 0 < t ≤ π 4 ,
and St ≥ S
π 4 Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves A generalization by Q.-Tang Pinkall-Thorbergsson Construction
Corollary (Q.-Tang, arXiv, 2018) (1). For the initial value F(·, 0) : M+ → S2l−1(1), F(x, y; 0) = ( √ 2 cos β(0)x, √ 2 sin β(0)y) with 0 < β(0) < π
4 , the mean curvature flow from F(·, 0) is given by
F : M+ × (−∞, T) → S2l−1(1), F(x, y; t) = ( √ 2 cos β(t)x, √ 2 sin β(t)y), where cos 2β(t) = cos 2β(0)e4(l−m−1)t and 1 = cos 2β(0)e4(l−m−1)T. (2). The mean curvature flow F(M+, t) has type I singularity at T. More precisely, there exists a constant C > 0 such that supF(M+,t)|B|2 ≤ C T − t, ∀t ∈ [0, T). (3). As t → T, F(M+, t) converges to Sl−1(1) = {(x, 0) ∈ R2l | |x| = 1}. (4). As t → −∞, F(M+, t) converges to M+.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
1
Introduction Isoparametric foliation Isoparametric foliation in unit spheres Miscellaneous properties
2
Related constructions A generalization by Q.-Tang Pinkall-Thorbergsson Construction
3
Curvature of leaves Ricci curvature sectional curvature
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Karcher[Ast´ erisque, 1988]µg=3, ∃ two different Einstein metrics
Q.-Tang-Yan[A.G.A.G., 2013]: g=4, for induced metric of focal submanifolds: 1). All the M− of OT-FKM-type are not Einstein; M+ of OT-FKM-type is Einstein if and only if it is diffeomorphic to Sp(2) in the homogeneous case with (m1, m2) = (4, 3). 2). (m1, m2) = (2, 2), the focal submanifold diffeomorphic to G2(R5) is Einstein, while the other one diffeomorphic to CP3 is not. 3). (m1, m2) = (4, 5), both focal submanifolds are not Einstein.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Karcher[Ast´ erisque, 1988]µg=3, ∃ two different Einstein metrics
Q.-Tang-Yan[A.G.A.G., 2013]: g=4, for induced metric of focal submanifolds: 1). All the M− of OT-FKM-type are not Einstein; M+ of OT-FKM-type is Einstein if and only if it is diffeomorphic to Sp(2) in the homogeneous case with (m1, m2) = (4, 3). 2). (m1, m2) = (2, 2), the focal submanifold diffeomorphic to G2(R5) is Einstein, while the other one diffeomorphic to CP3 is not. 3). (m1, m2) = (4, 5), both focal submanifolds are not Einstein.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Let G be a compact Lie group, and H a closed subgroup: Wang-Ziller[Iventiones Math., 1986]: ∃ 12-dimensional homogeneous space SU(4)/SU(2), admits no homogeneous Einstein metric. B¨
homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics. Theorem (Peng and Q., in preparation) For any homogeneous isoparametric foliation (G, Sn+1) with g = 4 and (m1, m2) = (6, 9), the isoparametric hypersurface and both of focal submanifolds admit at least one G-invariant Einstein metric. Problem 1: Classify the G-invariant Einstein metrics? Problem 2: For the case of non-homogeneous OT-FKM-type isoparametric foliations, do the isoparametric hypersurface and focal submanifolds admit natural Einstein metrics?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Let G be a compact Lie group, and H a closed subgroup: Wang-Ziller[Iventiones Math., 1986]: ∃ 12-dimensional homogeneous space SU(4)/SU(2), admits no homogeneous Einstein metric. B¨
homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics. Theorem (Peng and Q., in preparation) For any homogeneous isoparametric foliation (G, Sn+1) with g = 4 and (m1, m2) = (6, 9), the isoparametric hypersurface and both of focal submanifolds admit at least one G-invariant Einstein metric. Problem 1: Classify the G-invariant Einstein metrics? Problem 2: For the case of non-homogeneous OT-FKM-type isoparametric foliations, do the isoparametric hypersurface and focal submanifolds admit natural Einstein metrics?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Let G be a compact Lie group, and H a closed subgroup: Wang-Ziller[Iventiones Math., 1986]: ∃ 12-dimensional homogeneous space SU(4)/SU(2), admits no homogeneous Einstein metric. B¨
homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics. Theorem (Peng and Q., in preparation) For any homogeneous isoparametric foliation (G, Sn+1) with g = 4 and (m1, m2) = (6, 9), the isoparametric hypersurface and both of focal submanifolds admit at least one G-invariant Einstein metric. Problem 1: Classify the G-invariant Einstein metrics? Problem 2: For the case of non-homogeneous OT-FKM-type isoparametric foliations, do the isoparametric hypersurface and focal submanifolds admit natural Einstein metrics?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Let G be a compact Lie group, and H a closed subgroup: Wang-Ziller[Iventiones Math., 1986]: ∃ 12-dimensional homogeneous space SU(4)/SU(2), admits no homogeneous Einstein metric. B¨
homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics. Theorem (Peng and Q., in preparation) For any homogeneous isoparametric foliation (G, Sn+1) with g = 4 and (m1, m2) = (6, 9), the isoparametric hypersurface and both of focal submanifolds admit at least one G-invariant Einstein metric. Problem 1: Classify the G-invariant Einstein metrics? Problem 2: For the case of non-homogeneous OT-FKM-type isoparametric foliations, do the isoparametric hypersurface and focal submanifolds admit natural Einstein metrics?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Let G be a compact Lie group, and H a closed subgroup: Wang-Ziller[Iventiones Math., 1986]: ∃ 12-dimensional homogeneous space SU(4)/SU(2), admits no homogeneous Einstein metric. B¨
homogeneous spaces up to dimension 11 admit homogeneous Einstein metrics. Theorem (Peng and Q., in preparation) For any homogeneous isoparametric foliation (G, Sn+1) with g = 4 and (m1, m2) = (6, 9), the isoparametric hypersurface and both of focal submanifolds admit at least one G-invariant Einstein metric. Problem 1: Classify the G-invariant Einstein metrics? Problem 2: For the case of non-homogeneous OT-FKM-type isoparametric foliations, do the isoparametric hypersurface and focal submanifolds admit natural Einstein metrics?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Wallach[Ann. Math, 1972], Karcher[Ast´ erisque, 1988]µg = 3, each isoparametric hypersurface admits positively curved metrics. Observations for g = 4: 1). for the homogeneous case, the isoparametric hypersurface and focal submanifolds admit metrics with non-negative sectional curvature; 2). all isoparametric hypersurfaces and focal submanifolds admit almost nonnegatively curved metrics; 3). all isoparametric hypersurfaces and focal submanifolds are rationally elliptic. Problem(Tang): for g = 4, do any isoparametric hypersurface and focal submanifolds admit natural metrics with non-negative curvature?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Wallach[Ann. Math, 1972], Karcher[Ast´ erisque, 1988]µg = 3, each isoparametric hypersurface admits positively curved metrics. Observations for g = 4: 1). for the homogeneous case, the isoparametric hypersurface and focal submanifolds admit metrics with non-negative sectional curvature; 2). all isoparametric hypersurfaces and focal submanifolds admit almost nonnegatively curved metrics; 3). all isoparametric hypersurfaces and focal submanifolds are rationally elliptic. Problem(Tang): for g = 4, do any isoparametric hypersurface and focal submanifolds admit natural metrics with non-negative curvature?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Wallach[Ann. Math, 1972], Karcher[Ast´ erisque, 1988]µg = 3, each isoparametric hypersurface admits positively curved metrics. Observations for g = 4: 1). for the homogeneous case, the isoparametric hypersurface and focal submanifolds admit metrics with non-negative sectional curvature; 2). all isoparametric hypersurfaces and focal submanifolds admit almost nonnegatively curved metrics; 3). all isoparametric hypersurfaces and focal submanifolds are rationally elliptic. Problem(Tang): for g = 4, do any isoparametric hypersurface and focal submanifolds admit natural metrics with non-negative curvature?
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Considering g = 4, (m1, m2) = (8, 7), one of the OT-FKM-type isoparametric foliation on S31, Spin(9) × SO(2) acts on S31 and preserves leaves such that 1). Spin(9) × SO(2) acts on M+ transitively and M+ = (Spin(9) × SO(2))/(G2 × SO(2)) ∼ = Spin(9)/G2; 2). Spin(9) × SO(2) acts on M with cohomogeneity one and the group diagram Spin(9) × SO(2) ⊃ SU(3) × SO(2), G2 × Z2 ⊃ SU(3) × Z2; 3). Spin(9) × SO(2) acts on M− with cohomogeneity one and the group diagram Spin(9) × SO(2) ⊃ SU(4) × SO(2), Spin(7) × Z2 ⊃ SU(4) × Z2.
Chao Qian Geometric constructions related to isoparametric foliations
Introduction Related constructions Curvature of leaves Ricci curvature sectional curvature
Chao Qian Geometric constructions related to isoparametric foliations