Geometry of statistical submanifolds FURUHATA Hitoshi (Hokkaido - - PowerPoint PPT Presentation

Geometry of statistical submanifolds

FURUHATA Hitoshi (Hokkaido University) PADGE2012

Contents

• 1. What is a statistical manifold ?

— Basic Definitions and Examples

• 2. Spaces of constant Hessian curvature zero

— Elementary Submanifold Theory

• 3. Spaces of nonzero constant Hessian curvature

— Problems References.

Furuhata H. and Kurose T., Hessian manifolds of nonpositive constant Hessian sectional curvature, To appear in Tohoku Math. J. Furuhata H., Statistical hypersurfaces in the space of Hessian curvature zero, Differential Geom.

• Appl. 29(2011), S86–S90.

Furuhata H., Hypersurfaces in statistical manifolds, Differential Geom. Appl. 27(2009), 420–429.

1

What is a statistical manifold ?

M : C∞ manifold ∇ : torsion-free affine connection on M, g : Riemannian metric on M Definition. (∇, g) is a statistical structure on M

def

⇐ ⇒ ∇g ∈ Γ(TM (0,3)) is symmetric, i.e. (∇Xg)(Y, Z) = (∇Y g)(X, Z), ∀X, Y, Z ∈ Γ(TM). Definition. (M, ∇, g), ( M, ∇, g) : statistical manifolds (1) f : M → M is a statistical immersion

def

⇐ ⇒ g = f ∗ g and g(∇XY, Z) = g( ∇Xf∗Y, f∗Z), ∀X, Y, Z ∈ Γ(TM). (2) (M, ∇, g) = ( M, ∇, g)

def

⇐ ⇒ ∃ f : (M, ∇, g) → ( M, ∇, g) : bijective statistical immersion

2

What is a statistical manifold ?

M : C∞ manifold ∇ : torsion-free affine connection on M, g : Riemannian metric on M Definition. (∇, g) is a statistical structure on M

def

⇐ ⇒ ∇g ∈ Γ(TM (0,3)) is symmetric, i.e. (∇Xg)(Y, Z) = (∇Y g)(X, Z), ∀X, Y, Z ∈ Γ(TM). Definition. (M, ∇, g), ( M, ∇, g) : statistical manifolds (1) f : M → M is a statistical immersion

def

⇐ ⇒ g = f ∗ g and g(∇XY, Z) = g( ∇Xf∗Y, f∗Z), ∀X, Y, Z ∈ Γ(TM). (2) (M, ∇, g) = ( M, ∇, g)

def

⇐ ⇒ ∃ f : (M, ∇, g) → ( M, ∇, g) : bijective statistical immersion

2

Definition (cont.)

(3) f : (M, ∇, g) → ( M, ∇, g) : statistical hypersurface ξ ∈ Γ(f −1T M) : unit normal vector field of f Define h ∈ Γ(TM (0,2)), A∗ ∈ Γ(TM (1,1)), τ ∗ ∈ Γ(TM (0,1)) by

• ∇Xf∗Y = f∗∇XY + h(X, Y )ξ,
• ∇Xξ = −f∗A∗X + τ ∗(X)ξ,

∀X, Y ∈ Γ(TM). Remark. f : M → ( M, ∇, g) : immersion Set g := f ∗ g and ∇ by g(∇XY, Z) = g( ∇Xf∗Y, f∗Z). Then (∇, g) is called a statistical structure induced by f.

3

Notation. (1) ∇g : Levi-Civita connection of g (2) K := K(∇,g) := ∇ − ∇g ∈ Γ(TM (1,2)) Example. (M, g) : Riemannian manifold = ⇒ (∇g, g) is a statistical structure on M with K = 0. Remark. f : M → ( M, ∇, g) : immersion (∇, g) : statistical structure induced by f

• K := K(

∇, g) = 0 =

⇒ K := K(∇,g) = 0.

4

Example 1. (R+)n := {y = t(y1, . . . , yn) ∈ Rn : yi > 0} ∇n : affine connection determined by ∇n ∂ ∂yi ∂ ∂yj = −δij(yj)−1 ∂ ∂yj g0 :=

n

∑

j=1

(dyj)2 : Euclidean metric on (R+)n = ⇒ N n := ((R+)n, ∇n, g0) is a statistical manifold with K ̸= 0.

5

(M, ∇, g) : statistical manifold Definition. (∇, g) is a Hessian structure on M

def

⇐ ⇒ ∇ is flat ⇐ ⇒ Locally “ ∃φ : function on M st g = ∇dφ ”. Definition (Shima H., J.Math.Soc.Japan 1995). (M, ∇, g) : Hessian manifold, c ∈ R (∇, g) is of CHC c (i.e. of constant Hessian curvature c)

def

⇐ ⇒ (∇XK) (Y, Z) = −c 2 {g(X, Y )Z + g(X, Z)Y } , ∀X, Y, Z ∈ Γ(TM) ⇐ ⇒ The corresponding K¨ ahler metric on TM is of constant holomorphic sectional curvature −c. = ⇒ g is of constant curvature −c/4.

6

Spaces of CHC zero

Theorem 1 (F.-Kurose). (0) (Rn, ∇g0, g0) is a Hessian manifold of CHC zero. (1) N n = ((R+)n, ∇n, g0) is a Hessian manifold of CHC zero. (2) N k × (Rn−k, ∇g0, g0), k = 0, 1, . . . , n, are the only simply-connected Hessian manifolds of CHC zero with complete connection. (∇, g) is of CHC c

def

⇐ ⇒ (∇XK) (Y, Z) = −c 2 {g(X, Y )Z + g(X, Z)Y } , ∀X, Y, Z ∈ Γ(TM)

7

Spaces of CHC zero

Theorem 1 (F.-Kurose). (0) (Rn, ∇g0, g0) is a Hessian manifold of CHC zero. (1) N n = ((R+)n, ∇n, g0) is a Hessian manifold of CHC zero. (2) N k × (Rn−k, ∇g0, g0), k = 0, 1, . . . , n, are the only simply-connected Hessian manifolds of CHC zero with complete connection. Theorem 2. φ : N n → N n : statistical diffeo ⇐ ⇒ ∃σ ∈ Sn (: permutation on {1, . . . , n}) st φ : (R+)n ∋ (yi) → (yσ(i)) ∈ (R+)n

7

Spaces of CHC zero

Theorem 1 (F.-Kurose). (0) (Rn, ∇g0, g0) is a Hessian manifold of CHC zero. (1) N n = ((R+)n, ∇n, g0) is a Hessian manifold of CHC zero. (2) N k × (Rn−k, ∇g0, g0), k = 0, 1, . . . , n, are the only simply-connected Hessian manifolds of CHC zero with complete connection. Theorem 2. φ : N n → N n : statistical diffeo ⇐ ⇒ ∃σ ∈ Sn (: permutation on {1, . . . , n}) st φ : (R+)n ∋ (yi) → (yσ(i)) ∈ (R+)n

7

Theorem 3. f : N n → N n+1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: ∃L = (lα

j ) ∈ Mn+1 n(R), b = (bα) ∈ Rn+1 :

(1) lα

j = 0 or 1,

α = 1, . . . , n + 1, j = 1, . . . , n, (2) rank L = n, (3) lα

1 + · · · + lα n ≤ 1,

and f α((yi)) = ebα(y1)lα

1 · · · (yn)lα n.

Review. ξ : unit normal vector field of f,

• ∇Xf∗Y = f∗∇XY + h(X, Y )ξ,

∀X, Y ∈ Γ(TM).

8

Theorem 3. f : N n → N n+1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: ∃L = (lα

j ) ∈ Mn+1 n(R), b = (bα) ∈ Rn+1 :

(1) lα

j = 0 or 1,

α = 1, . . . , n + 1, j = 1, . . . , n, (2) rank L = n, (3) lα

1 + · · · + lα n ≤ 1,

and f α((yi)) = ebα(y1)lα

1 · · · (yn)lα n.

Example. L =   1 1   , b =   b3   ⇝ f(y) =   y1 y2 eb3  

8’

Theorem 3. f : N n → N n+1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: ∃L = (lα

j ) ∈ Mn+1 n(R), b = (bα) ∈ Rn+1 :

(1) lα

j = 0 or 1,

α = 1, . . . , n + 1, j = 1, . . . , n, (2) rank L = n, (3) lα

1 + · · · + lα n ≤ 1,

and f α((yi)) = ebα(y1)lα

1 · · · (yn)lα n.

Example. L =   1 1 1   , b =   log cos θ log sin θ   ⇝ f(y) =   y1 (cos θ)y2 (sin θ)y2   (θ ∈ (0, π/2))

9

Theorem 3. f : N n → N n+1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: ∃L = (lα

j ) ∈ Mn+1 n(R), b = (bα) ∈ Rn+1 :

(1) lα

j = 0 or 1,

α = 1, . . . , n + 1, j = 1, . . . , n, (2) rank L = n, (3) lα

1 + · · · + lα n ≤ 1,

and f α((yi)) = ebα(y1)lα

1 · · · (yn)lα n.

Outline of Proof.

• 1. h = 0 ⇝ the Euclidean second fundamental form of f vanishes.
• 2. Set f α((yi)) = ∑ aα

j yj + bα, and determine (aα j ) and (bα).

10

Problem. Construct and classify statistical immersions of N n into N n+1. Construct cylindrical statistical immersions as a first step. u, v : R+ ⊃ I → R+ : (u′(t))2 + (v′(t))2 = 1 (♮)1 R+ × I ∋ [s t ] =: x → f(x) :=   s u(t) v(t)   ∈ (R+)3 Proposition 4. (1) f : (R+ × I, ∇2, g0) → ((R+)3, ∇3, g0) is a statistical immersion ⇐ ⇒ (u′(t))3(u(t))−1 + (v′(t))3(v(t))−1 = t−1, t ∈ I (♮)2

11

Problem. Construct and classify statistical immersions of N n into N n+1. Construct cylindrical statistical immersions as a first step. u, v : R+ ⊃ I → R+ : (u′(t))2 + (v′(t))2 = 1 (♮)1 R+ × I ∋ [s t ] =: x → f(x) :=   s u(t) v(t)   ∈ (R+)3 Proposition 4. (1) f : (R+ × I, ∇2, g0) → ((R+)3, ∇3, g0) is a statistical immersion ⇐ ⇒ (u′(t))3(u(t))−1 + (v′(t))3(v(t))−1 = t−1, t ∈ I (♮)2

11

Proposition 4 (cont.)

(2) { u(t) = 3−3/2 { 3at2/3 + 2b }3/2 , v(t) = 3−3/2 1 − a3 −1 { 3(1 − a3)t2/3 − 2a2b }3/2 satisfy ODEs (♮)1 and (♮)2 for a ̸= 1, b ∈ R (due to HU Na). Remark. (1) b = 0 = ⇒ f is a plane (I = R+). (2) a < 0 and b > 0 = ⇒ f is the cylinder over a curve of the asteroid type. In fact, hu(t)2/3 + kv(t)2/3 = 1 (h, k = const > 0, t ∈ I ⊊ R+).

0.000 0.005 0.010 0.015 0.020 0.000 0.005 0.010 0.015 0.020

a = −2, b = 3/4

12

Proposition 4 (cont.)

(2) { u(t) = 3−3/2 { 3at2/3 + 2b }3/2 , v(t) = 3−3/2 1 − a3 −1 { 3(1 − a3)t2/3 − 2a2b }3/2 satisfy ODEs (♮)1 and (♮)2 for a ̸= 1, b ∈ R (due to HU Na). Question.

• 1. Prove they are the only solutions of (♮)1 and (♮)2.
• 2. Prove f is a plane if I = R+.

Solve u(t) > 0, v(t) > 0, (u′(t))2 + (v′(t))2 = 1, (♮)1 (u′(t))3(u(t))−1 + (v′(t))3(v(t))−1 = t−1. (♮)2

13

Proposition 4 (cont.)

(2) { u(t) = 3−3/2 { 3at2/3 + 2b }3/2 , v(t) = 3−3/2 1 − a3 −1 { 3(1 − a3)t2/3 − 2a2b }3/2 satisfy ODEs (♮)1 and (♮)2 for a ̸= 1, b ∈ R (due to HU Na). Question.

• 1. Prove they are the only solutions of (♮)1 and (♮)2.
• 2. Prove f is a plane if I = R+.

Solve u(t) > 0, v(t) > 0, (u′(t))2 + (v′(t))2 = 1, (♮)1 (u′(t))3(u(t))−1 + (v′(t))3(v(t))−1 = t−1. (♮)2 It was solved by Luc Vrancken after this talk.

13(add)

Spaces of CHC negative

Example 2. Ω := {1, 2, . . . , n + 1} ∋ x (finite sample space) ∆n := {η = t(η1, . . . , ηn) ∈ Rn : ηi > 0,

n

∑

l=1

ηl < 1} ∋ η p(x, η) :=      ηi, x = i ∈ {1, 2, . . . , n} 1 −

n

∑

l=1

ηl, x = n + 1 ⇝ p(·, η) is a positive probability density function on Ω parametrized by ∆n.

14

Example 2 (cont.)

∂i := ∂/∂ηi gF : Fisher information metric for ∆n ∋ η → p(·, η) gF

η (∂i, ∂j)

= ∑

x∈Ω

{∂i log p(x, η)}{∂j log p(x, η)}p(x, η) = · · · = (ηi)−1δij + (1 −

n

∑

l=1

ηl)−1. ∇(e) : exponential connection for ∆n ∋ η → p(·, η) gF

η (∇(e) ∂i ∂j, ∂k)

= ∑

x∈Ω

{∂i∂j log p(x, η)}{∂k log p(x, η)}p(x, η) = · · · = −(ηi)−2δijδjk − (1 −

n

∑

l=1

ηl)−2. = ⇒ ∆n := (∆n, ∇(e), gF) is a Hessian manifold of CHC −1.

15

Example 2 (cont.)

∂i := ∂/∂ηi gF : Fisher information metric for ∆n ∋ η → p(·, η) gF

η (∂i, ∂j)

= ∑

x∈Ω

{∂i log p(x, η)}{∂j log p(x, η)}p(x, η) = · · · = (ηi)−1δij + (1 −

n

∑

l=1

ηl)−1. ∇(e) : exponential connection for ∆n ∋ η → p(·, η) gF

η (∇(e) ∂i ∂j, ∂k)

= ∑

x∈Ω

{∂i∂j log p(x, η)}{∂k log p(x, η)}p(x, η) = · · · = −(ηi)−2δijδjk − (1 −

n

∑

l=1

ηl)−2. = ⇒ ∆n := (∆n, ∇(e), gF) is a Hessian manifold of CHC −1.

15

Theorem 5. The following f is a statistical immersion of ∆n into N n+1:

f(η) :=      2 √ p(1, η) . . . 2 √ p(n, η) 2 √ p(n + 1, η)      =          2 √ η1 . . . 2√ηn 2

• 1 −

n

∑

l=1

ηl          ∈ Sn(2) ∩ (R+)n+1

16

Theorem 6 (F.-Kurose). ∆n = (∆n, ∇(e), gF) is the only simply-connected Hessian manifold

• f CHC −1 with complete connection.

Theorem 7. φ : ∆n → ∆n : statistical diffeo ⇐ ⇒ ∃σ ∈ Sn+1 : φ : ∆n ∋ (ηi) → (ησ(i)) ∈ ∆n, where ηn+1 := 1 − ∑n

l=1 ηl.

Idea of Proof. Show the following to use Theorem 2: f : ∆n → N n+1 : statistical immersion as before ∀φ : ∆n → ∆n : statistical diffeo ∃ φ : N n+1 → N n+1 : statistical diffeo st f ◦ φ = φ ◦ f

17

Theorem 6 (F.-Kurose). ∆n = (∆n, ∇(e), gF) is the only simply-connected Hessian manifold

• f CHC −1 with complete connection.

Theorem 7. φ : ∆n → ∆n : statistical diffeo ⇐ ⇒ ∃σ ∈ Sn+1 : φ : ∆n ∋ (ηi) → (ησ(i)) ∈ ∆n, where ηn+1 := 1 − ∑n

l=1 ηl.

Idea of Proof. Show the following to use Theorem 2: f : ∆n → N n+1 : statistical immersion as before ∀φ : ∆n → ∆n : statistical diffeo ∃ φ : N n+1 → N n+1 : statistical diffeo st f ◦ φ = φ ◦ f

17

Theorem 6 (F.-Kurose). ∆n = (∆n, ∇(e), gF) is the only simply-connected Hessian manifold

• f CHC −1 with complete connection.

Theorem 7. φ : ∆n → ∆n : statistical diffeo ⇐ ⇒ ∃σ ∈ Sn+1 : φ : ∆n ∋ (ηi) → (ησ(i)) ∈ ∆n, where ηn+1 := 1 − ∑n

l=1 ηl.

Idea of Proof. Show the following to use Theorem 2: f : ∆n → N n+1 : statistical immersion as before ∀φ : ∆n → ∆n : statistical diffeo ∃ φ : N n+1 → N n+1 : statistical diffeo st f ◦ φ = φ ◦ f

17

Spaces of CHC positive

Review. Ω := (Ω, dx) ∋ x : measure space (sample space) M : C∞ manifold (parameter space) p : M ∋ θ → p(·, θ) ∈ {positive probability density on Ω} with adequate regular conditions ∂i := ∂/∂θi gF(∂i, ∂j) := ∫

Ω

{∂i log p(x, θ)}{∂j log p(x, θ)}p(x, θ)dx : Fisher information metric for p (assuming it is a Riemannian metric on M) gF

θ (∇(e) ∂i ∂j, ∂k) =

∫

Ω

{∂i∂j log p(x, θ)}{∂k log p(x, θ)}p(x, θ)dx Define ∇(m) of M by ∂igF(∂j, ∂k) = gF(∇(e)

∂i ∂j, ∂k) + gF(∂j, ∇(m) ∂i ∂k)

: mixture connection for p

18

= ⇒ (M, ∇(e), gF) and (M, ∇(m), gF) are statistical manifolds. Example 3. Ω := (R, dx) ∋ x, M := R × R+ ∋ θ = t(θ1, θ2) p(x, θ) := N(x, √ 2θ1, (θ2)2) := 1 √ 2π(θ2)2 exp { − 1 2(θ2)2(x − √ 2θ1)2 } : normal distribution of mean µ = √ 2θ1 ∈ R and variance σ2 = (θ2)2 > 0

4 2 2 4 0.05 0.10 0.15 0.20

19

= ⇒ (M, ∇(e), gF) and (M, ∇(m), gF) are statistical manifolds. Example 3. Ω := (R, dx) ∋ x, M := R × R+ ∋ θ = t(θ1, θ2) p(x, θ) := N(x, √ 2θ1, (θ2)2) := 1 √ 2π(θ2)2 exp { − 1 2(θ2)2(x − √ 2θ1)2 } : normal distribution of mean µ = √ 2θ1 ∈ R and variance σ2 = (θ2)2 > 0 ⇝ g−1 := 1

2gF = (θ2)−2 {

(dθ1)2 + (dθ2)2} ∇(m) : ∇(m)

∂1 ∂1 = 2(θ2)−1∂2, ∇(m) ∂2 ∂2 = (θ2)−1∂2, ∇(m) ∂1 ∂2 = ∇(m) ∂2 ∂1 = 0

19’

Theorem 8. (0) H2 := (R × R+, ∇(m), g−1) is a Hessian manifold of CHC 4. “(1)” H2 is the only 2-dim simply-connected Hessian manifold of CHC 4 with complete metric (due to Kurose). (2) φ : H2 → H2 : statistical diffeo ⇐ ⇒ ∃b, c ∈ R, c > 0 : φ1(y) = ±cy1 + b, φ2(y) = cy2. Problem. (1) Classify spaces of CHC positive. (2) Characterize the following (Hn, ∇(m), g−1). Hn := Rn−1 × R+(∋ y), ∇(m)

∂α ∂β = (yn)−1(2

∑

i,j

δi

αδj βδij + δn αδn β)∂n, g−1 := (yn)−2 ∑ α

(dyα)2

20

Theorem 8. (0) H2 := (R × R+, ∇(m), g−1) is a Hessian manifold of CHC 4. “(1)” H2 is the only 2-dim simply-connected Hessian manifold of CHC 4 with complete metric (due to Kurose). (2) φ : H2 → H2 : statistical diffeo ⇐ ⇒ ∃b, c ∈ R, c > 0 : φ1(y) = ±cy1 + b, φ2(y) = cy2. Problem. (1) Classify spaces of CHC positive. (2) Characterize the following (Hn, ∇(m), g−1). Hn := Rn−1 × R+(∋ y), ∇(m)

∂α ∂β = (yn)−1(2

∑

i,j

δi

αδj βδij + δn αδn β)∂n, g−1 := (yn)−2 ∑ α

(dyα)2

20

Theorem 8. (0) H2 := (R × R+, ∇(m), g−1) is a Hessian manifold of CHC 4. “(1)” H2 is the only 2-dim simply-connected Hessian manifold of CHC 4 with complete metric (due to Kurose). (2) φ : Hn → Hn : statistical diffeo ⇐ ⇒ ∃A = (ai

j) ∈ O(n − 1), b = (bi) ∈ Rn−1, c > 0 :

φi(y) = c ∑ ai

jyj + bi

(i, j = 1, . . . , n − 1), φn(y) = cyn. Problem. (1) Classify spaces of CHC positive. (2) Characterize the following (Hn, ∇(m), g−1). Hn := Rn−1 × R+(∋ y), ∇(m)

∂α ∂β = (yn)−1(2

∑

i,j

δi

αδj βδij + δn αδn β)∂n, g−1 := (yn)−2 ∑ α

(dyα)2

20

THE END

http://www.math.sci.hokudai.ac.jp/~furuhata/

Riemannian hypersurfaces

Theorem. ( M, ∇, g) : Hessian manifold of CHC c, (M, ∇, g) : Riemannian Hessian manifold i.e. K = 0 ∃ f : (M, ∇, g) → ( M, ∇, g) : statistical hypersurface = ⇒ c ≥ 0 Moreover,

• c > 0 =

⇒ (Riemannian shape operator S of f ) = ±1 2 √

• cI.

A11

Riemannian hypersurfaces (cont.)

Hn+1 = (Rn × R+, ∇(m), g−1) : “upper half” space of CHC 4 y0 > 0 fixed f0 : Rn ∋ t(y1, . . . , yn) → t(y1, . . . , yn, y0) ∈ Hn+1 Theorem. (M, ∇, g) : connected Riemannian Hessian manifold of dim n f : (M, ∇, g) → Hn+1 : statistical immersion = ⇒ f(M) is an open subset of f0(Rn)

A12

Hyperspheres

Theorem. f : (M n, ∇, g) → ((R+)n+1, ∇n+1, g0) : nondegenerate statistical hypersurface with n ≥ 3 (∇, g) is a Hessian structure of CHC −4r−2 < 0 = ⇒ f(M) is a part of the hypersphere of radius r and center 0. Outline of Proof. (1) (∇, g) is of CHC c = ⇒ g is of constant curvature −c/4. (2) f(M) is a part of a hypersphere of radius r. (3) The center c of the hypersphere is the origin.

A21

Proof of Theorem 1(2)

(1) (∇, g) is of CHC c = ⇒ g is of constant curvature −c/4. (2) (M, g) : Riemannian manifold of constant curvature κ K ∈ Γ(TM (1,2)), ∇ := ∇g + K (M, ∇, g) is a Hessian manifold of CHC −4κ ⇐ ⇒ K satisfies the following four conditions: (i) K(X, Y ) = K(Y, X), (ii) g(K(X, Y ), Z) = g(Y, K(X, Z)), (iii) κ{g(Y, Z)X − g(X, Z)Y } = K(Y, K(X, Z)) − K(X, K(Y, Z)), (iv) (∇gK)(X, Y ; Z) = 2κ{g(Z, X)Y + g(Z, Y )X} −K(Z, K(X, Y )) + K(K(Z, X), Y ) + K(X, K(Z, Y )).

A31

Proof of Theorem 1(2) (cont.)

(3) (M, g) : as above, o ∈ M fixed K0 ∈ ToM (1,2) satisfies the conditions (i)-(iii) at o = ⇒ ∃1K ∈ Γ(TM (1,2)) : (i) - (iv) on M and K0 = K(o). (4) Classify K0 satisfying (i) -(iii) and solve (iv) for K when κ = 0. ⇝ ∃k ∈ {0, 1, . . . , n} : K = −

n

∑

j=n−k+1

(yj)−1dyj ⊗ dyj ⊗ ∂ ∂yj ⇝ (M, ∇, g)

=

֒ → N k × Rn−k locally.

A32

Statistical manifolds and tangent bundles

M : manifold, π : TM → M : tangent bundle of M V : TM ∋ v → Vv := Ker(π∗)v ⊂ TvTM : vertical distribution ∇ : affine connection on M H : TM ∋ v → Hv ⊂ TvTM : horizontal distribution w.r.t. ∇ TvTM = Hv ⊕ Vv, v ∈ TM (π∗)v : Hv

=

− → Tπ(v)M ( ∃k∇

v : Vv =

− → Tπ(v)M ) J = J∇ ∈ Γ(TTM (1,1)) : J(XH ⊕ XV ) := (−XV ) ⊕ XH ⇝ J is an almost complex structure on TM

A41

Statistical manifolds and tangent bundles (cont.)

g : Riemannian metric on M G = G(∇,g) ∈ Γ(TTM (0,2)) : G(XH ⊕ XV , Y H ⊕ Y V ) := g(XH, Y H) + g(XV , Y V ) (1) (∇, g) is a statistical structure on M ⇐ ⇒ (J, G) is an almost K¨ ahler structure on TM. (2) (∇, g) is a Hessian structure on M ⇐ ⇒ (J, G) is a K¨ ahler structure on TM. (3) (∇, g) is a Hessian structure of CHC c ⇐ ⇒ (TM, J, G) is a K¨ ahler manifold

• f constant holomorphic sectional curvature −c.

A42

Hyperboloids and indefinite spaces

Problem. Construct an indefinite space of CHC 0 so that Hn = (Rn−1 × R+, ∇(m), g−1) is realized as a hyperboloid in it. Proposition. (1) Ln+1 := (Rn × R+, D, G0) is an indefinite complete space of CHC 0. (2) φ : Hn → Ln+1 is a statistical immersion.

ϕ : Hn ∋ x → (xn)−1          −x1 . . . −xn−1

1 √ 2

∑

j

(xj)2

1 √ 2

         ∈ Rn × R+ hyperboloid in the Lorentzian space

A51

Hyperboloids and indefinite spaces (cont.) i, j = 1, . . . , n − 1, α, β = 1, . . . , n, A, B = 1, . . . , n + 1

Define Ln+1 = (Rn × R+, D, G0) by G0 := ∑

j

(dyj)2 − 2dyndyn+1

D

∂ ∂yA

∂ ∂yB = (yn+1)−2 ∑

i

{ yiδA n+1δB n+1 − yn+1(δA iδB n+1 + δA n+1δB i) } ∂ ∂yi +(yn+1)−3  (yn+1)2    ∑

i,j

δAiδBjδij − (δAnδB n+1 + δA n+1δBn)    − ∑

i

yiyn+1(δAiδB n+1 + δA n+1δBi) +( ∑

j

(yj)2 + ynyn+1)δA n+1δB n+1   ∂ ∂yn −(yn+1)−1δA n+1δB n+1 ∂ ∂yn+1

A52

Hyperboloids and indefinite spaces (cont.)

Problem.

• 1. Characterize Ln+1.
• 2. Realize Hn in (∆N, ∇(e), gF) ⊂ ((R+)N+1, ∇N+1, g0).

(In particular, when n = 2. )

A53

The Gauss, Codazzi and Ricci equations

for a statistical hypersurface in a Hessian manifold

• ∇Xf∗Y = f∗∇XY + h(X, Y )ξ,
• ∇Xξ = −f∗A∗X + τ ∗(X)ξ.

(G) R∇(X, Y )Z = h(Y, Z)A∗X − h(X, Z)A∗Y, (C1) (∇Xh)(Y, Z) + τ ∗(X)h(Y, Z) = (∇Y h)(X, Z) + τ ∗(Y )h(X, Z), (C2) (∇XA∗)Y − τ ∗(X)A∗Y = (∇Y A∗)X − τ ∗(Y )A∗X, (R) h(X, A∗Y ) − h(Y, A∗X) = dτ ∗(X, Y ).

A61

Geodesics of spaces of CHC

Proposition. c = t(c1, c2) ∈ N 2, d = t(d1, d2) ∈ R2 γ : R ⊃ I → N 2 : the geodesic w.r.t. ∇2 st γ(0) = c and γ′(0) = d γ(t) = t(c1 exp(d1 c1t), c2 exp(d2 c2t))

A71