Almost Hermitian Structures on Tangent Vector Bundles

Almost Hermitian Structures on Tangent Vector Bundles

Hiroyasu Satoh

Graduate School of Pure and Applied Sciences, University of Tsukuba

E-mail: [email protected]

The 11th International Workshop on Differential Geometry November 9-11, 2006

Kyungpook National University

Table of Contents

1. Introduction

• ^Motivation

• Main Theorem

• ^Results

2. Definition and Proof of Main Theorem 3. Construction of examples:

• 1 parameter family of almost K ¨ahler structures on T M

• K ¨ahler Einstein structures on T M

Introduction

• ^Let (M, g) be a Riemannian manifold,

• π : T M → M the tangent bundle over M,

• D an affine connection on M.

T_ξ(T M) = H_ξ(T M) ⊕ V_ξ(T M) (ξ ∈ T_xM) ( H_ξ(T M), V_ξ(T M) ^isom= T_xM )







almost complex structure J^D =



 O I

−I O





Hermitian metric eg^D = g ⊕ g

Find the condition under which (J^D,eg^D) is almost K ¨ahler, (J^D,eg^D) is K ¨ahler,

eg^D is Einstein.

Motivation

• ^“The almost complex structure of a compact almost K¨ahler Einstein mani- fold is integrable.” (Goldberg ’69)

• ^{Let (J}, g) be an almost Hermitian structure on M.

(J, g) is almost K ¨ahler ⇐⇒^def dΩ = 0 (Ω = g(J·, ·))

(J, g) is K ¨ahler ⇐⇒^def dΩ = 0 and J is integrable

• s_g ≥ 0 =⇒ the Goldberg conjecture is true. (Sekigawa ’87)

• s_g < 0 =⇒ ^?

• ∃ strictly almost K ¨ahler Ricci flat space of dimension 4.

(Nurowski-Przanowski ’99)

Q. Is it possible to construct strictly almost K ¨ahler Einstein structures on T M? ( −→ × It is still under investigation.)

Main Theorem

1. ( J

, e g

) is almost K ¨ahler

⇐⇒

T

= 0 .

(i.e. D^∗ is the dual connection of D with respect to g⁾

2. ( J

, e g

) is K ¨ahler

⇐⇒

T

= T

= 0 , R

= R

= 0 .

(i.e. (g, D) is a Hessian structure on M)

3. e g

is an Einstein metric = ⇒ R

= 0 .

Construction of examples:

• 1 parameter family of almost K ¨ahler structures on T M

• K ¨ahler Einstein structures on T M

Known Results

• ^Let ∇ be the Levi-Civita connection of g^.

Then (J^∇,eg^∇) is an almost K ¨ahler structure on T M.

Moreover, (J^∇,eg^∇) is K ¨ahler ⇐⇒^iff R^∇ = 0. (Tachibana-Okumura ’62)

• e∇Re = 0 (i.e., (T M,eg^∇) is a loc. sym. space) =⇒ R^∇ = 0 =⇒ Re = 0. (Kowalski ’71)

• (T M,eg^∇) has constant scalar curvature ⇐⇒^iff R^∇ = 0. (Musso-Tricerri ’88)

• J^D is integrable ⇐⇒^iff D is a flat connection (R^D = 0^，T^D = 0).

(Dombrowski ’62)

Definition

• ^Let (x¹, . . . , xⁿ) be a local coordinate system on U ⊂ M,

• (x¹, . . . , xⁿ, y¹, . . . , yⁿ) a loc. coord. system on π⁻¹(U) ⊂ T M (yⁱ = dxⁱ).

• ^Let D be an affine connection on M,

• {Γ^k_{i j}} connection coefficients of D



 i.e. D ∂

∂xi

∂

∂xⁱ = X

k

Γ^k_{i j} ∂

∂x^k



^.

For X = X

i

Xⁱ ∂

∂xⁱ , ξ ∈ T_xM, we call X_ξ^H = X

i

Xⁱ ∂

∂xⁱ − X

i,j,k

Γ^k_{i j}Xⁱ y^j(ξ) ∂

∂y^{k the} horizontal lift of X at ξ^, X_ξ^V = X

i

Xⁱ ∂

∂y^{i the} vertical lift of X at ξ

Definition

We define the almost complex structure J^D and the Hermitian metric eg^{D by}

• J^D(X_ξ^H) = X_ξ^V, J^D(X_ξ^V) = −X_ξ^H

• eg^D(X_ξ^H, Y_ξ^H) = eg^D(X_ξ^V, Y_ξ^V) = g(X, Y), eg^D(X_ξ^H, Y_ξ^V) = 0.

We call (J^D,eg^D) the natural almost Hermitian structure on T M induced by (g, D).

• The dual connection D^∗ of D with respect to g is defined by

Z (g(X, Y)) = g(D_ZX, Y) + g(X, D^∗_ZY) ( X, Y, Z ∈ X(M) ).

• g(T^D∗(X, Y), Z) = (D_Xg)(Y, Z) − (D_Yg)(X, Z) + g(T^D(X, Y), Z),

• g(R^D∗(X, Y)Z, W) = −g(R^D(X, Y)W, Z).

Proof of Main Theorem:

• ^Let Ω = eg^D(J^D · , · ) be the K ¨ahler form of (J^D,eg^D). Then we have dΩ(X_ξ^H, Y_ξ^H, Z_ξ^H) = S

X,Y,Z g(R^D∗(X, Y)Z, ξ), dΩ(X_ξ^H, Y_ξ^H, Z_ξ^V) = g(T^D∗(X, Y), Z),

dΩ(X_ξ^H, Y_ξ^V, Z_ξ^V) = dΩ(X_ξ^V, Y_ξ^V, Z_ξ^V) = 0.

∴ dΩ = 0 ⇐⇒^iff T^D∗ = 0.

• ^Let Ω^∗ be the standard symplectic form on the cotangent bundle T^∗M and ϕ_g : T M → T^∗M the natural isomorphism induced by g^{. Then}

(J^D,eg^D) is almost K ¨ahler ⇐⇒^iff Ω = ϕ^∗_g(Ω^∗) (Here ϕ^∗_g(Ω^∗) is the pull-back of Ω^{∗ by} ϕ_g^.)

• From Dombrowski’s result,

(J^D,eg^D) is K ¨ahler ⇐⇒^iff T^D = T^D∗ = 0, R^D(= R^D∗) = 0.

Proof of Main Theorem:

Proposition. If the Ricci tensor Ric g of e g

^is J

-invariant or J

- anti-invariant, then R

= 0 .

(Outline of Proof)

Ric(Xg _ξ^H, Y_ξ^H) = ±Ric(Xg _ξ^V, Y_ξ^V)

Comparing the ξ-dependent parts of both sides of the above equation, we have R^D = 0.

Then, we have

eg^D is Einstein =⇒ Ricg is J^D-invariant =⇒ R^D = 0.

Example 1:

If (J^D,eg^D) is a strictly almost K ¨ahler structure on T M, then (g, D) satisfies

 R^D = 0, T^D = 0

T^D∗ , 0 (D^∗ is the dual conn. of D with respect to g⁾ · · · (∗)

• (M, g) = (S¹ × N, dθ² + h) where (N, h) is a space of positive constant curvature.

• ^Let ω = k dθ be an 1-form on M (k ∈ R),

• ^Let D be a torsion-free affine connection defined by Dg = ω ⊗ g^. Then, there exists k ∈ R ^s.t. R^D = 0. Fix k s.t. R^D = 0. For λ ∈ R^,

• g_λ = g + λ

|ω|²_g ω ⊗ ω

• ^Let D^∗_λ be the dual connection of D with respect to g_λ^.

Example 1:

• (g_λ, D^∗_λ) satisfies the condition (∗).

Then (g_λ, D^∗_λ) induces a family of almost K ¨ahler structures (J^D∗λ, ge_λ^D∗λ) on T M parametrized by λ. Moreover,

e

g_λ^D∗λ is Einstein ⇐⇒^iff λ = −2

=⇒ 

 g⁽₋²⁾ is pseudo-Riemannian D^∗₍₋₂₎ is torsion-free

=⇒ (T M, J^D∗(−2), gg⁽₋^2)D∗(−2)) is pseudo-K ¨ahler, Einstein

Q. Does there exists a pair (g, D) such that (J^D,eg^D) is strictly almost K ¨ahler Einstein structure ?

((J^D,eg^D) is almost K ¨ahler Einstein =⇒ J^D is integrable ?)

Example 2:

⇐⇒def D and D^∗ are flat connections.

⇐⇒ D is flat connection and ∃ϕ ∈ C^∞(M) s.t. g = Ddϕ^. We can define

• the 1st Koszul form α (α(X)dv_g = D_Xdv_g),

• the 2nd Koszul form β = Dα ^(symmetric (0, 2)-tensor field).

(M, g, D) is Hesse-Einstein ⇐⇒^def β = cg

⇐⇒iff (T M, J^D,eg^D) is K ¨ahler-Einstein

∵ ^If (g, D) is a Hessian structure on M, then the Ricci tensor Ricg of eg^D satisfies

Ric(Xg ^H, Y^H) = Ric(Xg ^V, Y^V) = −β(X, Y), Ric(Xg ^H, Y^V) = 0.

Example 2:

• ^Let Sn be the set of all symmetric n-matrices,

• S⁺_n ⊂ Sⁿ the set of all positive definite symmetric n-matrices.

• ^Let ρ : U → Sn be a injective linear map on U ⊂ R^{m (}ρ(U) ⊂ S⁺_n ^).

• ^For (µ, ξ) ∈ Rⁿ × U, p(x; µ, ξ) := q

detρ(ξ)

(2π)ⁿ exp n

−^{t(x−µ) ρ}₂^{(ξ) (x−µ)}o

(∈ C^∞(Rⁿ)).

• P^ρ = {p(x; µ, ξ) | (µ, ξ) ∈ R^m × U}. (Statistical model)

• ^Let D be the standard flat connection on P^ρ with respect to the coordi- nate (θ, ξ). (θ = ρ(ξ)µ⁾

• g_ρ = Ddϕ^. ϕ(θ, ξ) = 1

2{^tθρ(ξ)⁻¹θ − log det ρ(ξ)}

!

Then (g_ρ, D) is a Hessian structure on P^ρ.

Example 2:

• ρ : R₊ 3 t 7→ tI_n ∈ S^n.

• ^Let (g_ρ, D) be the Hessian structure on P^{ρ induced by} ρ^.

• ^Let D^∗ be the dual connection of D with respect to g_ρ^.

Then the natural almost Hermitian structure (J^D∗, ge_ρ^D∗) induced by (g_ρ, D^∗) has constant holomorphic sectional curvature (K ¨ahler-Einstein).

Q. How many Hessian structures which are Hesse-Einstein do there exist in the class of Hessian structures induced by matrix-valued injective linear maps?

Example 2:

Theorem. For any injective linear map ρ : U ⊂ R

→ S

^which satisfies ρ (U ) ⊂ S

, the Hessian structure ( g

, D

) on P

^{is al-} ways Hesse-Einstein. Hence (T P

, J

, g e

) is K ¨ahler-Einstein.

Here D

is the dual connection of D with respect to g

^.

(Outline of Proof)

Applying some suitable coordinate (linear) transformations, the injective lin- ear map ρ is reduced to the form

ρ(ξ¹, ξ²) = ξ¹ aξ¹ + bξ² aξ¹ + bξ² ξ²

!

(a, b ∈ R). Then, a direct computation shows

β^∗ = 3g .