. On totally geodesic surfaces in symmetric spaces and applications. Hiroshi Tamaru
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1 On totally geodesic surfaces in symmetric spaces and applications Hiroshi Tamaru Hiroshima University Workshop in honor of Professor Hiroo Naitoh s retirement 2016/March/05 Hiroshi Tamaru (Hiroshima University) 2016/March/05 1 / 28
2 Preface (1/2) Preface TG := totally geodesic many studies on TG-submfds in symmetric spaces Thanks Our studies are very influenced by Naitoh: a survey talk in Yuzawa 2009 Hiroshi Tamaru (Hiroshima University) 2016/March/05 2 / 28
3 Preface (2/2) Contents Main Result: 1: TG-surfaces in symmetric spaces Its applications: 2: TG-complex curves in Hermitian symm sp 3: TG-submfds in symmetric spaces of type AI Note This talk is based on joint works with Kentaro Kimura, Takayuki Okuda (Hiroshima U) Akira Kubo (Hiroshima Shudo U) Katsuya Mashimo (Hosei U) Hiroshi Tamaru (Hiroshima University) 2016/March/05 3 / 28
4 Introduction (1/3) Def (M, g) M is TG (totally geodesic) : [Second Fundamental Form] 0 γ : geodesic in M γ : geodesic in M Note Ẉe always assume that M is connected, complete Fundamental Problem For a given (irreducible) symmetric space (M, g), classify TG-submfds (up to isometric congruence) Hiroshi Tamaru (Hiroshima University) 2016/March/05 4 / 28
5 Introduction (2/3) Note Classifications of TG-submfds are known only for rk(m) = 1 by Wolf (1963), rk(m) = 2 by Klein ( ) Other cases would remain open Note It is hence natural to study particular TG-submfds: cplx (in Hermitian) by Satake (1965), Ihara (1967), reflective by Leung ( ), symmetric TG-submfds by Naitoh ( ), cf Chen-Nagano, Ikawa-Tasaki, Berndt-Olmos, Hiroshi Tamaru (Hiroshima University) 2016/March/05 5 / 28
6 Introduction (3/3) Our starting point Mashimo (cf Hashimoto et al) studies TG-surfaces: in M = G/K : symmetric space of cpt type, in terms of representations su(2) g What we thought We study TG-surfaces in M of noncpt type, the problem is essentially the same as the cpt case An advantage is one can use Iwasawa dec, solvable groups, Hiroshi Tamaru (Hiroshima University) 2016/March/05 6 / 28
7 TG-surfaces in symmetric spaces (1/7) Ṃ = G/K : irreducible symmetric sp of noncpt type Problem Classify TG-surfaces in M Problem (almost equivalent) Classify nonflat TG-surfaces in M Problem (almost equivalent) Classify nonabelian 2-dim LTS in p g = k p : the Cartan decomposition p V : Lie triple system : [[V, V ], V ] V Hiroshi Tamaru (Hiroshima University) 2016/March/05 7 / 28
8 TG-surfaces in symmetric spaces (2/7) Thm (primitive version) There is a correspondence between nonabelian 2-dim LTS in p, X n \ {0} satisfying (C1) [θx, X ] a + ; (C2) c > 0 : [[θx, X ], X ] = cx Notation θ : g g : the Cartan involution g = k a n : the Iwasawa decomposition a + := [positive closed Weyl chamber] Hiroshi Tamaru (Hiroshima University) 2016/March/05 8 / 28
9 TG-surfaces in symmetric spaces (3/7) Recall There is a correspondence between nonabelian 2-dim LTS in p, X n \ {0} satisfying (C1) [θx, X ] a + ; (C2) c > 0 : [[θx, X ], X ] = cx Proof ( ) For such X, LTS is Σ X := Span{[θX, X ], (1 θ)x } ( ) It follows from the congruency of a, a +, Hiroshi Tamaru (Hiroshima University) 2016/March/05 9 / 28
10 TG-surfaces in symmetric spaces (4/7) Simplest Case M = SL(n, R)/SO(n) everything is linear algebra: θx = t X, p = Sym 0 (n, R) n = {upper triangular}, a = {diagonal tr = 0}, a + = {diag(a 1,, a n ) a a 1 a n } Prop (Fujimaru-Kubo-T) In SL(n, R)/SO(n), up to isometric congruence, n = 3 exactly 2 nonflat TG-surfaces; n = 4 exactly 4 nonflat TG-surfaces Hiroshi Tamaru (Hiroshima University) 2016/March/05 10 / 28
11 TG-surfaces in symmetric spaces (5/7) Recall correspondence between nonabelian 2-dim LTS in p, X n \ {0} satisfying (C1), (C2) Note (general theory behind) Mostow (1955): nonabelian 2-dim LTS subalgebras sl(2, R) g Jacobson-Morozov theorem: such subalgebras nilpotent orbits in g Hiroshi Tamaru (Hiroshima University) 2016/March/05 11 / 28
12 TG-surfaces in symmetric spaces (6/7) Thm (sophisticated version) G := Isom(M) one-to-one correspondence between {nonflat TG-surfaces in M}/G; {nilpotent orbits Ad G (X ) of g}/{±1} Remark For nilpotent orbits, {Ad G 0(X )} is well studied {Ad G (X )} is understandable, for some M Hiroshi Tamaru (Hiroshima University) 2016/March/05 12 / 28
13 TG-surfaces in symmetric spaces (7/7) Cor Let M := SL(n, R)/SO(n) Then one-to-one correspondence between {nonflat TG-surface in M}/Isom(M) {partition of n} \ {[1 n ]} Ex n = 3: #{[3], [2, 1]} = 2 n = 4: #{[4], [3, 1], [2, 2], [2, 1, 1]} = 4 n = 5: #{[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1]} = 6 Hiroshi Tamaru (Hiroshima University) 2016/March/05 13 / 28
14 TG-complex curves (1/6) Topic of this section M : irr Hermitian symmetric space of noncpt type M M : TG-complex curve (ie, dim C M = 1, dim R M = 2, J-invariant) Thm (Kubo-Okuda-T) Let M be as above Then # ( {TG-cplx curves in M}/Isom(M) ) = rk(m) Hiroshi Tamaru (Hiroshima University) 2016/March/05 14 / 28
15 TG-complex curves (2/6) Recall # ( {TG-cplx curves in M}/Isom(M) ) = rk(m) Ex (M := CH n ) (1) rk(ch n ) = 1, (2) 1 TG-complex curve (up to Isom(M)) (TG-cplx submfds are CH n CH n 1 CH 1 ) Hiroshi Tamaru (Hiroshima University) 2016/March/05 15 / 28
16 TG-complex curves (3/6) Recall # ( {TG-cplx curves in M}/Isom(M) ) = rk(m) Ex (M := G 2 (Rn ), n 4) (1) rk(g2 (Rn )) = 2, (2) two TG-complex curves (up to Isom(M)): M G2 (R4 ) = CH 1 CH 1 : TG-complex submfd TG-complex curves are: CH 1 {pt}, and diagonal CH 1 Hiroshi Tamaru (Hiroshima University) 2016/March/05 16 / 28
17 TG-complex curves (4/6) Step 1 (cf Satake (1966), Hermann (1962)) M : Hermitian, r := rk(m) M (TG-complex submfd) : M = (CH 1 ) r ( ) By taking the strongly orthogonal roots Step 2 (construction) One can construct r TG-maps φ : CH 1 (CH 1 ) r, The image lives in k {1, 2,, r} components Note: Their sectional curvatures are different Hiroshi Tamaru (Hiroshima University) 2016/March/05 17 / 28
18 TG-complex curves (5/6) Step 3 (they exhaust all) M M : TG-complex curve Then X n : T o M = Span{[θX, X ], (1 θ)x } Since T o M is J-invariant, we have X is in a good position (ie, (1 θ)x J(a)) By looking at the root spaces, we conclude M {previous examples} Note that, in particular, M (CH 1 ) r Hiroshi Tamaru (Hiroshima University) 2016/March/05 18 / 28
19 TG-complex curves (6/6) Recall # ( {TG-cplx curves in M}/Isom(M) ) = rk(m) Comment M M : TG-complex curve Then M (CH 1 ) r is actually known by Satake (1966) We determined the isometry classes Key Tool (recall) correspondence between nonabelian 2-dim LTS in p, X n \ {0} satisfying (C1), (C2) Hiroshi Tamaru (Hiroshima University) 2016/March/05 19 / 28
20 TG-submfds in AI (1/7) In this section, we propose a procedure to classify TG-submfds, and apply it to SL(n, R)/SO(n) with n = 3, 4 Procedure (Step 1) Classify all nonflat TG-surfaces Σ in M (This is a topic of the previous sections) (Step 2) For each Σ, classify nonflat TG-submfds ( Σ) Key Fact nonflat TG-submfd contains nonflat TG-surface Hiroshi Tamaru (Hiroshima University) 2016/March/05 20 / 28
21 TG-submfds in AI (2/7) Thm (Klein, cf Kimura) max TG-submfd in SL(3, R)/SO(3) is congruent to [SL(2, R)/SO(2)] R +, or SO 0 (1, 2)/S(O(1) O(2)) Note RH 2 = SL(2, R)/SO(2) = SO 0 (1, 2)/S(O(1) O(2)) Hiroshi Tamaru (Hiroshima University) 2016/March/05 21 / 28
22 TG-submfds in AI (3/7) Step 1 of Proof (Fujimaru-Kubo-T) exactly 2 nonflat TG-surfaces in SL(3, R)/SO(3): L 1 := Span 1 0, , L 2 := Span 1 0, Hiroshi Tamaru (Hiroshima University) 2016/March/05 22 / 28
23 TG-submfds in AI (4/7) Step 2 of Proof Consider L 1 (ToDo: Classify LTS L ( L 1 )): [L 1, L 1 ] =: X L must be ad X -invariant ad X -weight space dec: p L 1 = V 1 (0) V 2 (±i) Candidates: L = L 1 V 1 (0) or L 1 V 2 (±i) The former is LTS, but the latter is not Hiroshi Tamaru (Hiroshima University) 2016/March/05 23 / 28
24 TG-submfds in AI (5/7) Step 2 of Proof (Continued) Consider L 2 (ToDo: Classify LTS L ( L 2 )): By similar calculations, such L Thm (recall) max TG-submfd in SL(3, R)/SO(3) is congruent to [SL(2, R)/SO(2)] R +, or SO 0 (1, 2)/S(O(1) O(2)) Comment Ḅoth TG-submfds are reflective Hiroshi Tamaru (Hiroshima University) 2016/March/05 24 / 28
25 TG-submfds in AI (6/7) Thm (Kimura) max TG-submfd in SL(4, R)/SO(4) is congruent to [SL(3, R)/SO(3)] R +, Sp(2, R)/U(2), [SL(2, R)/SO(2)] [SL(2, R)/SO(2)] R +, SO 0 (2, 2)/S(O(2) O(2)), SO 0 (1, 3)/S(O(1) O(3)) Note Ṭhis would be a new result (rk (SL(4, R)/SO(4)) = 3) Hiroshi Tamaru (Hiroshima University) 2016/March/05 25 / 28
26 TG-submfds in AI (7/7) Proof (Step 1) Recall: exactly 4 nonflat TG-surfaces (Step 2) Classify TG-submfds containing one of them Cor M = SL(n, R)/SO(n) with n = 3, 4, M M : max TG-submfd M is reflective Question Why? What happens for n 5? Hiroshi Tamaru (Hiroshima University) 2016/March/05 26 / 28
27 Summary and Problems Our Studies TG-surfaces in symmetric spaces TG-complex curves in Hermitian symmetric spaces TG-submfds in SL(n, R)/SO(n) with n = 3, 4 Further Problems # ( {nonflat TG-surfaces in M}/Isom(M) ) =? Classify TG-submfds in SL(n, R)/SO(n) with n 5 Classify TG-submfds for other M Which M satisfies maximal TG reflective? Hiroshi Tamaru (Hiroshima University) 2016/March/05 27 / 28
28 Thank you! Congratulations on your retirement, and Wishing you a future filled with happiness!! Hiroshi Tamaru (Hiroshima University) 2016/March/05 28 / 28
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