On totally geodesic surfaces in symmetric spaces and applications
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1 On totally geodesic surfaces in symmetric spaces and applications Hiroshi Tamaru ( ) Hiroshima University The second China-Japan geometry conference, Fuzhou 2016/September/08
2 Preface (1/2) Preface TG := totally geodesic. many studies on TG-submfds in symmetric spaces. Claim many open problems on TG-submfds in symmetric spaces!
3 Contents Preface (2/2) 1: Introduction 2: TG-surfaces in symmetric spaces 3: TG-complex curves in Hermitian symm. sp. 4: TG-submfds in symmetric spaces of type AI Note This talk is based on some joint works with Takuya Fujimaru (Usuki High-School) Kentaro Kimura (Hokuryo High-School) Akira Kubo (Hiroshima Shudo U.) Katsuya Mashimo (Hosei U.) Takayuki Okuda (Hiroshima U.)
4 Introduction (1/4) Def. (M, g) M is TG (totally geodesic) : [Second Fundamental Form] 0 γ : geodesic in M γ : geodesic in M. Note We always assume that M is connected, complete. Fundamental Prop. TG-submfds in symmetric spaces Lie triple systems. duality between TG-submfds in cpt/noncpt types.
5 Introduction (2/4) What are known: Classifications of TG-submfds M in symm. spaces M are know for M flat; rk(m) = 1 (Wolf 1963); M irr. Hermitian, M cplx (Satake 1965, Ihara 1967); M irr., M reflective (Leung ); M irr., M polars/meridians (Chen-Nagano 1970 s);... M irr., rk(m) = 2, M maximal (Klein ).
6 Introduction (3/4) What are unknown: Classification of (maximal) TG-submfds for rk(m) 3; The index i(m) for many M, where i(m) := min codim{m : proper TG-submfd in M}; Classification when M reducible; Classification of ALL TG-submfds even when rk(m) = 2. Note Most cases study maximal TG-submfds. However, it is not easy to classify all TG-submfds from it.
7 Introduction (4/4) Our Strategy (1) We start from TG-surfaces (smallest nontrivial case). Contrast to: maximal TG-submfds are studied in many cases. TG-surfaces would be building blocks of all TG-submfds. Our Strategy (2) We assume that the ambient spaces M are of noncpt type. Because of the duality, essentially same as the cpt case. An advantage: one can use Iwasawa dec., solvable groups,...
8 TG-surfaces in symmetric spaces (1/7) M = 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 : LTS (Lie triple system) : [[V, V ], V ] V.
9 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].
10 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 +,...
11 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. 2014) In SL(n, R)/SO(n), up to isometric congruence, n = 3 exactly 2 nonflat TG-surfaces; n = 4 exactly 4 nonflat TG-surfaces.
12 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.
13 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 (in progress).
14 TG-surfaces in symmetric spaces (7/7) Cor. Let M := SL(n, R)/SO(n). Then one-to-one correspondence between Ex. {nonflat TG-surface in M}/Isom(M) {partition of n} \ {[1 n ]}. 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.
15 TG-complex curves (1/4) Topic of this section M : irr. Hermitian symmetric space of noncpt type. M M : TG-complex curve. (i.e., 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).
16 Recall TG-complex curves (2/4) # ( {TG-cplx curves in M}/Isom(M) ) = rk(m). Proof (Step 1) M : TG-surface in M X n satisfying (C1), (C2). (Step 2, cf. Satake) M : TG-cplx curve X is in a good position. (Step 3) One can describe such X s explicitly, in terms of roots. ( exactly r := rk(m) such X s)
17 TG-complex curves (3/4) 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 ).
18 TG-complex curves (4/4) Recall # ( {TG-cplx curves in M}/Isom(M) ) = rk(m). Ex. (M := G 2 (R n ), n > 4) (1) rk(g 2 (Rn )) = 2, (2) two TG-complex curves (up to Isom(M)): M G 2 (R4 ) = CH 1 CH 1 : TG-complex submfd. TG-complex curves are: CH 1 {pt}, and diagonal CH 1.
19 TG-submfds in AI (1/6) 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.
20 TG-submfds in AI (2/6) Thm. (Klein, cf. Kimura-Kubo-Okuda-T.) 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)).
21 TG-submfds in AI (3/6) Step 1 of Proof (Fujimaru-Kubo-T.) exactly 2 nonflat TG-surfaces in SL(3, R)/SO(3): L 1 := Span 0, 0 0 0, L 2 := Span 0,
22 TG-submfds in AI (4/6) 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.
23 TG-submfds in AI (5/6) 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 Both TG-submfds are reflective.
24 TG-submfds in AI (6/6) Thm. (Kimura-Kubo-Okuda-T.) 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 This must be the first classification result for a rank 3 space.
25 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. Determine the index i(m) := min codim[proper TG-submfds].
26 Thank you!
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