(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like

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1 () Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ) g ( ) = (M, g, ) ([4]) (M, g, ) M X, Y, Z, W (.) g((x, Y )Z, W ) = g(z, (X, Y )W ) (X, Y )Z = [ X, Y ]Z [X,Y ] Z (.3) (X, Y )Z = k{ g(y, Z)X g(x, Z)Y } (M, g, ) k 1

2 iemann (M, g) p(x; ξ) n M = { p(x; ξ) ξ = (ξ 1,..., ξ n ) Ξ n } (ξ 1,..., ξ n ) n iemann l = l(x; ξ) = log p(x; ξ) i = / ξ i p(x; ξ) E M g (.4) g ij = E[ i l j l ] (ξ 1,..., ξ n ) g Fisher E[ i l ] = 0 Fisher (.5) g ij = E[ i j l ] α [ ( Γ (α) ij,k = E i j l + 1 α ) ] (.6) i l j l k l α (α) (.7) g( (α) i j, k ) = Γ (α) ij,k α Fisher ( α) (α) (M, g, (α) ) (0) Fisher Levi Civita α (M, g, (α) ) α χ C F 1,..., F n n Θ φ [ ] (.8) p(x; θ) = exp C(x) + θ s F s (x) φ(θ) n θ = (θ 1,..., θ n ) Dirichlet Cauchy Weibull (.8) χ p(x; θ) dx = 1 [ ] (.9) exp φ(θ) = exp C(x) + θ s F s (x) dx. χ θ i i φ exp φ = exp φ E[ F i ] (.10) E[ F i ] = i φ i = / θ i j ( i φ exp φ) = exp φ E[ F i F j ] k {( i j φ + i φ j φ) exp φ} = exp φ E[ F i F j F k ] (.11) (.1) E[ F i F j ] = i j φ + i φ j φ, E[ F i F j F k ] = i j k φ + i j φ k φ + j k φ i φ + k i φ j φ + i φ j φ k φ

3 (.8) l(x; θ) = C(x) + θ s F s (x) φ(θ) (.5) i j l = i j φ Fisher g (.13) g ij = i j φ. g g 1 = (g ij ) (.6), (.10) (.13) (.14) Γ (α) ij,k = 1 (1 α) ig jk = 1 (1 α) i j k φ (.7) α (.15) (α) i j = 1 (1 α) sg ij g st t α (α) (.16) (α) ( i, j ) k = c(α) 4 ( j g ks i g st i g ks j g st ) t. c(α) = (1 α)(1 + α) ([1]) A (.16) ±1 3 1 (3.1) p(x; µ, σ) = ] 1 (x µ) exp [ π σ σ < x < µ σ Fisher (3.) M 1 = { (µ, σ) < µ <, σ > 0 } = + ds 1 = dµ + dσ σ µ = / µ, σ = / σ α (α) (3.3) (α) µ µ = 1 α σ σ, (α) µ σ = (α) σ µ = 1 + α σ (α) σ σ = 1 + α σ. σ 3 µ,

4 α (3.4) (α) ( µ, σ ) µ = c(α) σ σ, (α) ( µ, σ ) σ = c(α) σ µ (M 1, ds 1, (α) ) α c(α) ±1 Fisher (3.) Poincaré M = { (x, y) y > 0 } = + (3.5) ds = dx + dy y Poincaré (M, ds ) (Levi Civita ) 1 (Levi Civita ) x y y > 0 M 1 α α d µ (1 + α) dµ dσ dt σ dt dt = 0, (3.6) d σ dt + 1 α ( ) dµ 1 + α σ dt σ (1) σ σ > 0 ( ) dσ = 0 dt () α < 1 µ σ > 0 ( α = 1 µ σ > 0 ) (3) α = 1 µ σ > 0 µ (4) α > 1 µ σ > 0 Poincaré Poincaré M n+1 = { (x 1,..., x n, x n+1 ) x n+1 > 0 } = n + (3.7) ds = 1 (x n+1 ) { (dx 1 ) + + (dx n ) + (dx n+1 ) } Poincaré 1 n [ 1 (3.8) p(x; ξ) = ( π) n det Σ exp 1 4 ] t (x µ)σ 1 (x µ)

5 x = t (x 1,..., x n ) n, µ = t (µ 1,..., µ n ) n Σ = (σ ij ) ξ = (µ 1,..., µ n, σ 11, σ 1,..., σ 1n, σ,..., σ n,..., σ nn ) M = { ξ n+ 1 n(n+1) < µ i < (i = 1,..., n), (σ ij ) } Σ = diag (σ,..., σ ) (3.9) p(x; ξ) = 1 ( π σ) n n i=1 [ exp (x i µ i ) σ ξ = (µ 1,..., µ n, σ) M n+1 1 = { (µ 1,..., µ n, σ) < µ i < (i = 1,..., n), σ > 0 } = n + Fisher ] (3.10) ds 1 = 1 σ ( dµ dµ n + n dσ ) n = 1 1 Fisher n dµ dµ n n f(µ 1,..., µ n ) = n + dσ σ + g(σ) = 1 σ (M1 n+1, ds 1 ) doubly warped product Poincaré (Mi n+1, ds i ) (i = 1, ) ω (3.11) ds = 1 σ ( dµ dµ n + ω dσ ) M n+1 = { (µ 1,..., µ n, σ) < µ i < (i = 1,..., n), σ > 0 } = n + ω = 1 Poincaré (M n+1, ds ) ω = n (M1 n+1, ds n+1 1 ) M 1 α α (M n+1, ds ) (α) (3.1) (α) i j = 1 α ω σ ij σ, (α) i σ = (α) σ i = 1 + α σ (α) σ σ = 1 + α σ. σ i = / µ i, σ = / σ (M n+1, ds, (α) ) α c(α) ω ±1 (M n+1, ds, (α) ) α d µ i (1 + α) dµ i dσ dt σ dt dt = 0 (3.13) d σ dt + 1 α ( ) dµs ω 1 + α σ dt σ i, (i = 1,..., n), ( ) dσ = 0 dt 1 5

6 4 M J = I (1,1) J I M iemann (M, J) M X, Y g(jx, JY ) = g(x, Y ) (M, g, J) Hermite (4.1) g(jx, Y ) + g(x, J Y ) = 0 (1,1) J iemann (M, g) (M, g, J) Hermite-like (J ) = J (J ) = I (4.) g(jx, J Y ) = g(x, Y ) Hermite-like J (M, g,, J) Kähler-like (4.1) X, Y, Z (4.3) g(( Z J)X, Y ) + g(x, ( ZJ )Y ) = 0 ([6]) B (1) (M, g, J) Hermite-like (M, g, J ) Hermite-like () (M, g,, J) Kähler-like (M, g,, J ) Kähler-like Kähler M n M n > 1 M Kähler-like (X, Y )JZ = J(X, Y )Z (.3) Kähler ([11]) C Kähler-like (M n, g,, J) M (n > 1) M (M, g, (α) ) α J (α) J (α) M (4.1) (4.4) (J (α) ) = g 1 J (α) g α J (α) ( (α) i J (α)) { j = i J (α) t j + 1 ( ) } (1 α) J (α) r j i g rs g st i g js g sr J r (α) t t, (α) J (α) = 0 i J (α) k j + 1 ( (4.5) (1 α) j J (α) r ) i g rs g sk i g js g sr J r (α) k = 0. 6

7 α = 1 (4.6) P k j (4.7) J (1) k j = P k j Pj rp r i = j i J ( 1) k j ([11]) D = P s r g sj g rk (1) (M, g, J (±1) ) Hermite-like () (M, g, (±1), J (±1) ) Kähler-like (M, g, (α) ) α (.15) A (α) l ijk = c(α)a(g jk l i g ik l j ) A C ([11]) E (M n, g, (α) ) (n > 1) A 0 M (J (α) ) = I (4.5) α = ±1 5 1 ( 1) 3 Poincaré (M, ds, (α) ) (α) c(α) ω dim M = n (> ) E α = ±1 (±1) J (±1) ( j µ i + P i 1 j 4µ i s µ s J (1) σ = j σ s µ s + S J ( 1) = i µ j P j i ( 1 ω σ 4µ j s µ s + Λ = P j i p j sµ s Sµ j Q j ) P i j, Qi, j, S Λ = I j Q i S P i sµ s + Sµ i + Q i ) ω i σ, s µ s S 7

8 (1) (M, ds, J (±1) ) Hermite-like () (M, ds, (±1), J (±1) ) Kähler-like ω = 1 M Poincaré Poincaré Kähler Poincaré e m 4 (5.1) p(x; ξ) = Γ(m + x x n ) Γ(m) x 1!x! x n! p m 0 p x 1 1 pxn n ξ = (p 1,..., p n ) Γ(x) m k = 1,..., n x k {0, 1,,... } p k (> 0) p 0 + p p n = 1 { p(x; ξ) = exp log Γ(m + x x n ) log Γ(m) log x s! + } x s log p s + m log(1 p 1 p n ) C(x) = log Γ(m + x x n ) log Γ(m) log x s!, F i (x) = x i, θ i = log p i (i = 1,,..., n), φ(θ) = m log(1 p 1 p n ) M n = { (θ 1,..., θ n ) 0 < θ i < (i = 1,..., n) } = ( + ) n p i = e θi (i = 1,,..., n) (5.) τ(θ) = 1 e θs φ(θ) = m log τ(θ). (5.3) (5.4) (5.5) i φ = m e θi τ(θ), i j φ = m { i j k φ = m e θi e θj τ(θ) ij + e θi τ(θ) { e θi } τ(θ) ij ik + e θi τ(θ) } + e θi e θj e θk τ(θ) 3, e θk ij + e θj e θk τ(θ) ik + e θi e θj τ(θ) jk 8

9 i = / θ i (.13) (5.4) Fisher (5.6) g ij = m Fisher g (5.7) { g ij = e θi e θj τ(θ) ij + e θi τ(θ) τ(θ) m e θi ( ij e θi ) (.14), (5.5), (5.7) : { (5.8) Γ (α) k ij = Γ (α) ij,s gsk = 1 (1 α) α α (α) { (5.9) (α) i j = 1 (1 α) } ij ik + e θj τ(θ) ik + e θi τ(θ) jk ij i + e θj τ(θ) i + e θi τ(θ) j. } }. α [{ } { (α) ( i, j ) k = c(α) e θj 4 τ(θ) jk + e θj e θk e θi τ(θ) i τ(θ) ik + e θi τ(θ) e θk } j ] ([10]) F c(α) 4m (4.6), (4.7), (5.6), (5.7) 1 ( 1) J (1) J ( 1) J (1) k j = Pj k { ( ) } e θs J ( 1) k j = e θj e θk P j k e θk r=1 ([10]) P j r + 1 τ(θ) P s k e θk G dim M ( 4) (1) (M, g, J (±1) ) Hermite-like () (M, g, (±1), J (±1) ) Kähler-like r=1 P s r n = J (0) 1 1 = J (0) = ± J (0) 1 = 1 e θ e θ J (0) 1 = ± 1 e θ1 e θ1 ( ( ( e θ1 θ 1 e θ1 e θ e θ1 θ 1 e θ1 e θ e θ1 θ 1 e θ1 e θ ) 1 ) 1 ) 1,, (M, g, (0), J (0) ) Kähler 9

10 6 (M n, g) p M J p : T p M T p M ( ) ( ) ( ) ( ) J p x α = p y α, J p p y α = p x α (α = 1,..., n) p u, v T p M J p (u + 1 v) = J p u + 1 J p v J p : T C p M T C p M Z α = Zᾱ = z α = 1 ( x α 1 z α = z α = 1 ) y α, ( x α + 1 ) y α JZ α = 1 Z α, JZᾱ = 1 Zᾱ (α = 1,..., n) J = I M g p M g T p M u + 1 v, u + 1 v T C p M g(u + 1 v, u + 1 v ) = {g(u, u ) g(v, v )} + 1{g(u, v ) + g(v, u )} g T C p M (z 1,..., z n ) α, β = 1,,..., n g (z 1,..., z n ) g αβ, g α β, gᾱβ, gᾱ β g αβ = g(z α, Z β ), gᾱβ = g(zᾱ, Z β ), g α β = g(z α, Z β), gᾱ β = g(zᾱ, Z β) g αβ = g βα, gᾱ β = g βᾱ, g α β = g βα, ḡ αβ = gᾱ β, ḡ α β = gᾱβ M g u, v T p M g(ju, Jv) = g(u, v) g M Hermite g Hermite g g αβ = 0 (gᾱ β = 0) (α, β = 1,..., n) J M g u, v T p M (6.1) g(ju, J v) = g(u, v) 10

11 J J (M, g, J) Hermite-like (6.) (6.3) J Zβ = 1 ( g βσ g σω g β σ g σω) Z ω 1 ( g βσ g σ ω g β σ g σ ω) Z ω, J Z β = 1 ( g βσ g σω g β σ g σω) Z ω 1 ( g βσ g σ ω g β σ g σ ω) Z ω (J ) = I J Z β = JZ β ( 1 g βσ g σω Z ω + g σ ω ) (6.4) Z ω (6.5) = JZ β + ( 1 g g σω β σ Z ω + g σ ω ) Z ω, J Z β = JZ β ( 1 g βσ g σω Z ω + g σ ω ) Z ω = JZ β + 1 g β σ ( g σω Z ω + g σ ω Z ω ) 6.1 J = J g αβ = 0 (gᾱ β = 0) JJ Z α = ( g ασ g σω g α σ g σω) Z ω ( g ασ g σ ω g α σ g σ ω) Z ω, JJ Zᾱ = ( gᾱσ g σω gᾱ σ g σω) Z ω ( gᾱσ g σ ω gᾱ σ g σ ω) Z ω, J JZ α = ( g ασ g σω g α σ g σω) Z ω + ( g ασ g σ ω g α σ g σ ω) Z ω, J JZᾱ = ( gᾱσ g σω gᾱ σ g σω) Z ω ( gᾱσ g σ ω gᾱ σ g σ ω) Z ω JJ J J 6. [J, J ] = 0 ( ) g ασ g σ β = 0, g g σ β α σ = 0. gᾱσ g σβ = 0, gᾱ σ g σβ = JJ + J J = I g ασ g σβ = 0, g α σ g σβ = β α. ( gᾱσ g σ β = β ) ᾱ, gᾱ σ g σ β = 0. α (6.6) J (α) = 1 + α J + 1 α J 6.4 g ασ g σβ = 0, g g σβ α σ = α β α J (α) 7 A, B, C, = 1,..., n, 1,..., n ZA Z B = Γ E AB Z E, Z A Z B = Γ E AB Z E Z A = / z A Z A g(z B, Z C ) = g( ZA Z B, Z C ) + g(z B, Z A Z C ) Z A g BC = Γ E AB g EC + Γ E AC g BE 11

12 (7.1) (7.) (7.3) (7.4) (7.5) (7.6) Z α g βγ = Γ ε αβ g εγ + Γ ε αβ g εγ + Γ ε αγ g βε + Γ ε αγ g β ε, Z α g β γ = Γ ε αβ g ε γ + Γ ε αβ g ε γ + Γ ε α γ g βε + Γ ε α γ g β ε, Z α g βγ = Γ ε α β g εγ + Γ ε α β g εγ + Γ ε αγ g βε + Γ ε αγ g β ε, Z α g β γ = Γ ε α β g ε γ + Γ ε α β g ε γ + Γ ε α γ g βε + Γ ε α γ g β ε, Zᾱg βγ = Γ ε ᾱβ g εγ + Γ ε ᾱβ g εγ + Γ ε ᾱγ g βε + Γ ε ᾱγ g β ε, Zᾱg β γ = Γ ε ᾱβ g ε γ + Γ ε ᾱβ g ε γ + Γ ε ᾱ γ g βε + Γ ε ᾱ γ g β ε, (7.7) (7.8) Zᾱg βγ = Γ ε ᾱ β g εγ + Γ ε ᾱ β g εγ + Γ ε ᾱγ g βε + Γ ε ᾱγ g β ε, Zᾱg β γ = Γ ε ᾱ β g ε γ + Γ ε ᾱ β g ε γ + Γ ε ᾱ γ g βε + Γ ε ᾱ γ g β ε ( X J)Y = X (JY ) J X Y ( Zα J)Z β = 1 Γ ε αβ Z ε, ( Zα J)Z β = 1 Γ ε α β Z ε, ( ZᾱJ)Z β = 1 Γ ε ᾱβ Z ε, ( ZᾱJ)Z β = 1 Γ ε ᾱ β Z ε. 7.1 J Γ ε αβ = 0, ΓAβ ε = 0, Γ ε = 0 β A Γ ε α β = 0, Γ ε ᾱβ = 0, Γ ε ᾱ β = 0. J = 0 (7.1) (7.8) : (7.9) (7.10) (7.11) (7.1) (7.13) (7.14) (7.15) (7.16) (7.9) (7.11) (7.17) (7.18) Z α g βγ = Γ ε αβ g εγ + Γ ε αγ g βε + Γ ε αγ g β ε, Z α g β γ = Γ ε αβ g ε γ + Γ ε α γ g βε + Γ ε α γ g β ε, Z α g βγ = Γ ε αγ g βε + Γ ε αγ g β ε, Z α g β γ = Γ ε α γ g βε + Γ ε α γ g β ε, Zᾱg βγ = Γ ε ᾱγ g βε + Γ ε ᾱγ g β ε, Zᾱg β γ = Γ ε ᾱ γ g βε + Γ ε ᾱ γ g β ε, Zᾱg βγ = Γ ε ᾱ β g εγ + Γ ε ᾱγ g βε + Γ ε ᾱγ g β ε, Zᾱg β γ = Γ ε ᾱ β g ε γ + Γ ε ᾱ γ g βε + Γ ε ᾱ γ g β ε. Γ γ αβ + Γ ε αωg εβ g ωγ = g γω Z α g ωβ + g γ ω Z α g ωβ, Γ γ αβ + Γ ε αωg εβ g ω γ = g γω Z α g ωβ + g γ ω Z α g ωβ 1

13 (7.1) (7.13) (7.19) (7.0) Γ γ α β = gγ ω Z α g ω β + g γω Z βg ωα, Γ γ α β = g γ ω Z α g ω β + g γω Z βg ωα (7.14) (7.16) (7.1) (7.) Γ γ ε + Γ ᾱ β ᾱ ωg ε βg ωγ = g γω Zᾱg ω β + g γ ω Zᾱg ω β, Γ γ ε + Γ ᾱ β ᾱ ωg ε βg ω γ = g γω Zᾱg ω β + g γ ω Zᾱg ω β, Kähler-like J = J 6.1 J = J g αβ = 0 ( gᾱ β = 0 ) (7.9) (7.16) (7.3) (7.4) (7.5) (7.6) (7.7) (7.8) (7.9) (7.30) Γ ε αγ g β ε = 0, Z α g β γ = Γ ε αβ g ε γ + Γ ε α γ g β ε, Z α g βγ = Γ ε αγ g βε, Γ ε α γ g βε = 0, Γ ε ᾱγ g β ε = 0, Zᾱg β γ = Γ ε ᾱ γ g β ε, Zᾱg βγ = Γ ε ᾱ β g εγ + Γ ε ᾱγ g βε, Γ ε ᾱ γ g βε = 0 (7.3) 0 = Γαγ ε g β ε g β ω = Γ αγ ε ( ε ω g β ε g β ω ) = Γαγ ω Γ ω αγ = 0 = 0, Γ ᾱγ ω = 0 (7.5) Z αg βγ g βω = Γ ε (7.6), (7.7), (7.30) Γ ω α γ Γ ω Aγ Γ αγ ε (ε ω g βε g βω ) = Γ ω αγ = 0, Γ ω A γ Γ ω αγ (7.8) Γ ω ᾱ γ Γ ω ᾱγ = Γ ω γᾱ = 0, Γ ω α γ Γαβ εg ε γg γω = Γαβ ε( ε ω = Γ ω γα = Z α g εγ g εω = 0, Γ ᾱ γ ω = 0 αγ g βε g βω = = Zᾱg ε γ g ε ω = 0 (7.4) Z α g β γ = Γαβ εg ε γ Z α g β γ g γω = g εγ g γω ) = Γ ω αβ Γ ω αβ = Z αg β ε g εω (7.9) Γ ω ᾱ β = Z ᾱg βε g ε ω ΓAB C = Γ AB C 7.1 Kähler-like (M n, g,, J) J = J = 13

14 (7.31) (7.3) (7.33) (7.34) (7.35) (7.36) (7.37) (7.38) (7.39) (7.40) Z α g βγ Z β g αγ = Γ ε αγ g βε + Γ ε αγ g β ε Γ ε βγ g αε Γ ε βγ g α ε, Z α g βγ Z γ g βα = Γ ε αβ g εγ Γ ε γβ g εα, Z α g β γ Z β g α γ = Γ ε α γ g βε + Γ ε α γ g β ε Γ ε β γ g αε Γ ε β γ g α ε, Z α g β γ Z γ g βα = Γ ε αβ g ε γ, Z α g βγ Z γ g βα = 0, Z γ g βα Z α g β γ = Γ ε γ β g εα, Zᾱg β γ Z γ g βᾱ = 0, Zᾱg βγ Z βgᾱγ = Γ ε ᾱγ g βε + Γ ε ᾱγ g β ε Γ ε βγ g ᾱε Γ ε βγ g ᾱ ε, Zᾱg β γ Z βgᾱ γ = Γ ε ᾱ γ g βε + Γ ε ᾱ γ g β ε Γ ε β γ g ᾱε Γ ε β γ g ᾱ ε, Zᾱg β γ Z γ g βᾱ = Γ ε ᾱ β g ε γ Γ ε γ β g εᾱ 7. J (7.41) (7.4) (7.43) (7.44) Z α g σω + g σε Γ ω αε Z α g σ ω + g σε Γ ω αε Zᾱg σω + g σε Γ ω ᾱε Zᾱg σ ω + g σε Γ ω ᾱε + g σ ε Γ ω α ε = 0, + g σ ε Γ ω α ε = 0, + g σ ε Γ ω ᾱ ε = 0, + g σ ε Γ ω ᾱ ε = 0. Nijenhuis N(X, Y ) = [JX, JY ] J[X, JY ] J[JX, Y ] [X, Y ], N (X, Y ) = [J X, J Y ] J [X, J Y ] J [J X, Y ] [X, Y ] Kähler-like N(X, Y ) = 0, N (X, Y ) = 0 8 ABC D = Z A ΓBC D Z B ΓAC D + ΓBC E ΓAE D ΓAC E ΓBE D ABγ = Z A Γ Bγ Z B Γ Aγ + Γ E Bγ Γ AE Γ E Aγ Γ BE = Γ ε BγΓ Aε + Γ ε BγΓ A ε Γ ε AγΓ Bε Γ ε AγΓ B ε = 0, AB γ = Z A ΓB γ Z B ΓA γ + Γ E B γ ΓAE ΓA γ E ΓBE = Γ ε B γγ Aε + Γ ε B γγ A ε Γ ε A γγ Bε Γ ε A γγ B ε = 0. 14

15 ABγ = Z A ΓBγ Z B ΓAγ + Γ E Bγ ΓAE ΓAγ E ΓBE = Z A ΓBγ Z B ΓAγ + ΓBγΓ ε Aε + ΓBγΓ ε A ε ΓAγΓ ε Bε ΓAγΓ ε B ε = Z A ΓBγ Z B ΓAγ + ΓBγΓ ε Aε ΓAγΓ ε Bε, AB γ = Z A Γ B γ Z B Γ A γ + Γ E B γ Γ AE Γ E A γ Γ BE = Z A Γ B γ Z B Γ A γ + ΓB γγ ε Aε + ΓB γγ ε A ε ΓA γγ ε Bε ΓA γγ ε B ε = Z A Γ B γ Z B Γ A γ + ΓB γγ ε A ε ΓA γγ ε B ε 8.1 (8.1) (8.) (8.3) (8.4) (8.5) (8.6) (8.7) (8.8) ABγ = 0, AB γ = 0, αβγ ᾱβγ ᾱ = Z αγ βγ Z βγ αγ + Γ ε βγ Γ αε Γ ε αγ Γ βε, = Z ᾱγ βγ, βγ = 0, αβ γ = 0, α β γ ᾱ β γ = Z αγ β γ, = Z ᾱγ β γ Z β Γ ᾱ γ + Γ ε β γ Γ ᾱ ε Γ ε ᾱ γ Γ β ε. 8. ic ic αβ = εαβ ε, ic α β = ε εα β = Z αγ ε ε β, icᾱβ = εᾱβ ε = Z ᾱγ ε icᾱ β = ε εᾱ β. εβ, ABCD = g((z A, Z B )Z C, Z D ), ABCD = g( (Z A, Z B )Z C, Z D ) (.) ABCD = ABDC (8.9) ABC D F = ABE g F C g ED 8.3 αβc D = ω D α βc = Z β Γ ω αε αβε g ωcg εd, D ω ᾱ βc = ᾱ β ε g ωcg εd. g ωc g εd Z α Γ ω β ε g ωcg εd, 15

16 ic ic AB = DAB D = DABE ged ic AB = DABEg DE = (ABDE + BDAE)g DE = ABED g ED + DBAEg DE = ( BEAD + EABD )g ED + ic BA = ic BA ic AB + ic BA ic AB ic BA = ic AB + ic BA 8.4 ic ic 8.5 ic ic αβ = εασ ω g ωβ g σε Z ε Γασ ω g ωβ g σ ε + Z α Γ ε σ ω g ωβ g σ ε, ic α β = εασ ω g ω βg σε Z ε Γασ ω g ω βg σ ε + Z α Γ ε σ ω g ω βg σ ε, ic ᾱβ = Z ᾱγ ω εσ ic ᾱ β = Z ᾱγ ω εσ g ωβ g σε Z ε Γᾱ σ ω g ωβ g σε g ω βg σε Z ε Γᾱ σ ω g ω βg σε εᾱ σ ω g ωβ g σ ε, εᾱ σ ω g ω βg σ ε. r = ic AB g AB, r = ic AB gab r = ic AB g AB = DABE g DE g AB = ADEB gab g DE = ic DE gde = r 8.6 r = r 9 Kähler-like Kähler-like (M, g,, J) (9.1) ( A, B ) C = c 4 [ g( B, C ) A g( A, C ) B g( B, J C )J A + g( A, J C )J B +{g( A, J B ) g(j A, B )}J C ].. (9.1) J J (9.1) Kähler-like (M, g,, J) J = J (M, g,, J) c (9.1) 0 (9.) (9.3) αβγ α βγ = c (g βγ α g αγ β ), = c (g βγ α + g α β γ ), 16

17 (9.4) (9.5) α β γ ᾱ β γ = c (g α γ β = c + g α β γ ), (g β γ ᾱ gᾱ γ β ) (8.4) (8.7) (9.6) (9.7) Z βγ αγ = c (g βγ α + g βα γ ), Z α Γ β γ = c (g α γ β + g α β γ ) 8. (9.8) (9.9) (9.10) (9.11) ic βγ = c (n 1)g βγ, ic α β = c (n + 1)g α β, icᾱβ = c (n + 1)g ᾱβ, ic β γ = c (n 1)g β γ 9. (9.1) Kähler-like r (9.1) r = c{n(n + 1) g εω g εω }. 8.5 (9.), (9.5) (9.7) ic (9.13) (9.14) (9.15) (9.16) ic αβ = c {(n 1)g αβ 4g α ε g g ε ω β ω }, ic α β = c {(n + 1)g α β 4g αεg βω g εω }, ic ᾱβ = c {(n + 1)g ᾱβ 4gᾱε g βω g εω }, ic ᾱ β = c {(n 1)g ᾱ β 4g ᾱεg βω g εω } r (9.17) r = c{n(n + 1) 6g εω g εω + 4g αε g βω g αβ g εω } (9.18) r r = 4c(g εω g εω g αε g βω g αβ g εω ) (9.1) Kähler-like c = 0 g εω g εω g αε g βω g αβ g εω = 0 (g ε ω g ε ω gᾱ ε g β ω gᾱ βg ε ω = 0) 17

18 A = (g αβ ), B = (g αβ ) g εω g εω g αε g βω g αβ g εω = 0 tr (AB) tr (AB) = 0 tr (E AB) = tr (E AB) E [1] S. Amari, Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, 8 Springer-Verlag, [] S. Amari and H. Nagaoka, Methods of Information Geometry, AMS & Oxford University Press, 000. [3] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley & Sons, [4] M. Noguchi, Geometry of statistical manifolds, Differential Geom. Appl., (199), 197. [5] K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge Univ. Press, Cambridge, [6] K. Takano, Statistical manifolds with almost complex structures and its statistical submersions, Tensor N. S., 65 (004), [7], Examples of the statistical submersion on the statistical model, Tensor N. S., 65 (004), [8], Statistical manifolds with almost contact structures and its statistical submersions, J. Geom., 85 (006), [9], Geodesics on statistical models of the multivariate normal distribution, Tensor N. S., 67 (006), [10], Examples of statistical manifolds with almost complex structures, Tensor N. S., 69 (008), [11], Exponential families admitting almost complex structures, SUT J. Math., 46 (010), 1 1. [1] K. Yano and M. Kon, Structures on Manifolds, World Scientific, [13] [14]

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys