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. ( ∇ X g )( 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 ( � ⇒ g = f ∗ � ⇐ g and g ( ∇ X Y, Z ) = � ∇ X f ∗ 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. ( ∇ X g )( 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 ( � ⇒ g = f ∗ � ⇐ g and g ( ∇ X Y, Z ) = � ∇ X f ∗ 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 − 1 T � M ) : unit normal vector field of f Define h ∈ Γ( TM (0 , 2) ) , A ∗ ∈ Γ( TM (1 , 1) ) , τ ∗ ∈ Γ( TM (0 , 1) ) by � ∇ X f ∗ Y = f ∗ ∇ X Y + h ( X, Y ) ξ, � ∇ X ξ = − f ∗ A ∗ X + τ ∗ ( X ) ξ, ∀ X, Y ∈ Γ( TM ) . Remark . f : M → ( � M, � ∇ , � g ) : immersion g ( � Set g := f ∗ � g and ∇ by g ( ∇ X Y, Z ) = � ∇ X f ∗ 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 g ) = 0 = ⇒ K := K ( ∇ ,g ) = 0 . K := K ( � � ∇ , � 4

Example 1 . ( R + ) n := { y = t ( y 1 , . . . , y n ) ∈ R n : y i > 0 } ∇ n : affine connection determined by ∂y j = − δ ij ( y j ) − 1 ∂ ∂ ∇ n ∂ ∂y j ∂y i n ∑ ( dy j ) 2 : Euclidean metric on ( R + ) n g 0 := j =1 ⇒ N n := (( R + ) n , ∇ n , g 0 ) 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 ) ⇒ ( ∇ X K ) ( Y, Z ) = − c def ⇐ 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) ( R n , ∇ g 0 , g 0 ) is a Hessian manifold of CHC zero. (1) N n = (( R + ) n , ∇ n , g 0 ) is a Hessian manifold of CHC zero. (2) N k × ( R n − k , ∇ g 0 , g 0 ) , k = 0 , 1 , . . . , n , are the only simply-connected Hessian manifolds of CHC zero with complete connection. ( ∇ , g ) is of CHC c ⇒ ( ∇ X K ) ( Y, Z ) = − c def ⇐ 2 { g ( X, Y ) Z + g ( X, Z ) Y } , ∀ X, Y, Z ∈ Γ( TM ) 7

Spaces of CHC zero Theorem 1 (F.-Kurose). (0) ( R n , ∇ g 0 , g 0 ) is a Hessian manifold of CHC zero. (1) N n = (( R + ) n , ∇ n , g 0 ) is a Hessian manifold of CHC zero. (2) N k × ( R n − k , ∇ g 0 , g 0 ) , 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 ∈ S n (: permutation on { 1 , . . . , n } ) st φ : ( R + ) n ∋ ( y i ) �→ ( y σ ( i ) ) ∈ ( R + ) n 7

Spaces of CHC zero Theorem 1 (F.-Kurose). (0) ( R n , ∇ g 0 , g 0 ) is a Hessian manifold of CHC zero. (1) N n = (( R + ) n , ∇ n , g 0 ) is a Hessian manifold of CHC zero. (2) N k × ( R n − k , ∇ g 0 , g 0 ) , 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 ∈ S n (: permutation on { 1 , . . . , n } ) st φ : ( R + ) n ∋ ( y i ) �→ ( y σ ( i ) ) ∈ ( R + ) n 7

Theorem 3 . f : N n → N n +1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: j ) ∈ M n +1 n ( R ) , b = ( b α ) ∈ R n +1 : ∃ L = ( l α (1) l α j = 0 or 1 , α = 1 , . . . , n + 1 , j = 1 , . . . , n , (2) rank L = n , (3) l α 1 + · · · + l α n ≤ 1 , and f α (( y i )) = e b α ( y 1 ) l α 1 · · · ( y n ) l α n . Review. ξ : unit normal vector field of f, � ∇ X f ∗ Y = f ∗ ∇ X Y + 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: j ) ∈ M n +1 n ( R ) , b = ( b α ) ∈ R n +1 : ∃ L = ( l α (1) l α j = 0 or 1 , α = 1 , . . . , n + 1 , j = 1 , . . . , n , (2) rank L = n , (3) l α 1 + · · · + l α n ≤ 1 , and f α (( y i )) = e b α ( y 1 ) l α 1 · · · ( y n ) l α n . Example .       y 1 1 0 0 y 2   , b =   ⇝ f ( y ) =   0 1 0 L = e b 3 b 3 0 0 8’

Theorem 3 . f : N n → N n +1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: j ) ∈ M n +1 n ( R ) , b = ( b α ) ∈ R n +1 : ∃ L = ( l α (1) l α j = 0 or 1 , α = 1 , . . . , n + 1 , j = 1 , . . . , n , (2) rank L = n , (3) l α 1 + · · · + l α n ≤ 1 , and f α (( y i )) = e b α ( y 1 ) l α 1 · · · ( y n ) l α n . Example .       y 1 1 0 0   , b =   ⇝ f ( y ) =   (cos θ ) y 2 0 1 log cos θ L = (sin θ ) y 2 0 1 log sin θ ( θ ∈ (0 , π/ 2) ) 9

Theorem 3 . f : N n → N n +1 : statistical hypersurface h = 0 = ⇒ f is a hyperplane expressed as follows: j ) ∈ M n +1 n ( R ) , b = ( b α ) ∈ R n +1 : ∃ L = ( l α (1) l α j = 0 or 1 , α = 1 , . . . , n + 1 , j = 1 , . . . , n , (2) rank L = n , (3) l α 1 + · · · + l α n ≤ 1 , and f α (( y i )) = e b α ( y 1 ) l α 1 · · · ( y n ) l α n . Outline of Proof . 1. h = 0 ⇝ the Euclidean second fundamental form of f vanishes. 2. Set f α (( y i )) = ∑ a α j y j + b α , and determine ( a α j ) and ( b α ) . 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   [ s ] s R + × I ∋   ∈ ( R + ) 3 =: x �→ f ( x ) := u ( t ) t v ( t ) Proposition 4 . (1) f : ( R + × I, ∇ 2 , g 0 ) → (( R + ) 3 , ∇ 3 , g 0 ) 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   [ s ] s R + × I ∋   ∈ ( R + ) 3 =: x �→ f ( x ) := u ( t ) t v ( t ) Proposition 4 . (1) f : ( R + × I, ∇ 2 , g 0 ) → (( R + ) 3 , ∇ 3 , g 0 ) 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.) { u ( t ) = 3 − 3 / 2 { } 3 / 2 , 3 at 2 / 3 + 2 b (2) v ( t ) = 3 − 3 / 2 � � 1 − a 3 � � − 1 { } 3 / 2 3(1 − a 3 ) t 2 / 3 − 2 a 2 b satisfy ODEs ( ♮ ) 1 and ( ♮ ) 2 for a ̸ = 1 , b ∈ R (due to HU Na). Remark . ⇒ f is a plane ( I = R + ). (1) b = 0 = (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 + ). 0.020 0.015 0.010 0.005 a = − 2 , b = 3 / 4 0.000 0.000 0.005 0.010 0.015 0.020 12