Examples

It is well-known that all irreducible symmetric K¨ahler manifolds of non-compact type have nonpositive sectional curvature. In this chapter, we study the curvature operator of the following classical type irreducible symmetric K¨ahler manifolds of noncompact type.

1 Type I

_m,n

Let M be an open subset D^I_m,n = ©

ζ ∈M(m, n;C) ¯

¯I_n−^tζζ >0ª

of C^mn, where I_n is the n×n identity matrix, ^tζ is the transpose of ζ ∈ M and ζ denotes the complex conjugate ofζ ∈M.

Let ζ = (z_ip) ∈ M with z_ip = x_ip +√

−1y_ip, i = 1, . . . , m and p = 1, . . . , n be the canonical complex coordinate system of M. Let Φ be a real-valued function in a coordinate neighborhoodU at the origin 0 of M defined by

Φ(ζ) = log det¡

I_n−^tζζ¯ ¢₋₁

=X

|z_ip|²+ 1 2

Xz¯_ipz_iqz¯_jqz_jp+ (higher order terms)

for any ζ ∈U.

Unless otherwise stated, Greek indices α, β, . . . denote all subscripts appearing as pairs{11,12, . . . , mn}, while Latin capitalsA, B, . . .denote{11,12, . . . , mn,11, . . . , mn}.

We set

Z_α = ∂

∂z = 1 2

µ ∂

∂x −√

−1 ∂

∂y

¶ ,

Z_α = ∂

∂z¯_α = 1 2

µ ∂

∂x_α +√

−1 ∂

∂y_α

¶ . The K¨ahler metric g of M is given, at the origin 0, by

g_αβ = ∂²Φ

∂zα∂zβ

(0), g_αβ¯= ∂²Φ

∂z_α∂z¯_β(0) ¡

=g_βα¢ , g_αβ = ∂²Φ

∂z¯_α∂z¯_β(0), whereg_AB =g(Z_A, Z_B)(0). Note that we have

g_{ip jq} =g_{jq ip}=δ_ijδ_pq, g_αβ =g_αβ = 0.

LetR be the curvature tensor of the K¨ahler manifold (M, g), and define RABCD = g(R(ZC, ZD)ZB, ZA)(0). Then we obtain

Rip jq kr ls = ∂⁴Φ

∂z_ip∂z_jq∂z_kr∂z_ls(ζ)

¯¯

¯¯

ζ=0

=δijδklδpsδqr+δilδjkδpqδrs.

By the symmetry properties of R, it is easy to see that R_ABCD =R_CDAB =−R_BACD and R_αβCD =R_αβCD = 0.

Then it is immediate to see that the curvature operator ˆR of (M, g) is given by R(Zˆ ip∧Z_jq) = −δij

Xm

k=1

Zkp∧Z_kq−δpq

Xn

r=1

Zir∧Z_jr, R(Zˆ _α∧Z_β) = ˆR(Z_α∧Z_β) = 0.

We divide our investigation into the following four cases: (1)m =n = 1; (2)m=n = 2;

(3) m=n≥3; (4) m 6=n.

(1) In this case, ˆR has two eigenvalues 0 and −2, whose eigenvectors are the fol-lowing:

0 ; Z11∧Z11, Z₁₁∧Z₁₁,

−2 ; Z₁₁∧Z₁₁.

(2) In this case, ˆR has three eigenvalues 0,−2 and−4, whose eigenvectors are given as follows:

0 ; Z₁₁∧Z₁₁−Z₁₂∧Z₁₂−Z₂₁∧Z₂₁+Z₂₂∧Z₂₂, Z₁₁∧Z₂₁−Z₁₂∧Z₂₂, Z₂₁∧Z₁₁−Z₂₂∧Z₁₂, Z₁₁∧Z₁₂−Z₂₁∧Z₂₂, Z₁₂∧Z₁₁−Z₂₂∧Z₂₁, Z₁₁∧Z₂₂, Z₁₂∧Z₂₁, Z₂₁∧Z₁₂, Z₂₂∧Z₁₁, Z_α∧Z_β, Z_α∧Z_β, for any α, β,

−2 ; Z₁₁∧Z₁₁−Z₂₂∧Z₂₂, Z₁₂∧Z₁₂−Z₂₁∧Z₂₁, Z₁₁∧Z₂₁+Z₁₂∧Z₂₂, Z₂₁∧Z₁₁+Z₂₂∧Z₁₂, Z₁₁∧Z₁₂+Z₂₁∧Z₂₂, Z₁₂∧Z₁₁+Z₂₂∧Z₂₁,

−4 ; Z₁₁∧Z₁₁+Z₁₂∧Z₁₂+Z₂₁∧Z₂₁+Z₂₂∧Z₂₂.

(3) In this case, ˆR has three eigenvalues 0,−m and −2m, whose eigenvectors are given respectively by

0 ; Zip∧Z_ip−Zim∧Z_im−Zmp∧Zmp−Zmm∧Zmm for i, p6=m, Z_kp∧Z_kq−Z_mp∧Z_mq for k 6=m and p6=q,

Zir∧Z_jr−Zim∧Z_jm for r6=m and j 6=i, Z_ip∧Z_jq for i6=j and p6=q,

Zα∧Zβ, Zα∧Z_β for any α, β,

−m ;

Xm

r=1

Z_ir∧Z_ir− 1 m−2

Ã_m−1 X

k=1 m−1X

r=1

Z_kr∧Z_kr+Z_mm∧Z_mm

!

for i6=m, Xm

k=1

Z_kp∧Z_kp− 1 m−2

ÃXm k=1

Xm

r=1

Z_kr∧Z_kr+Z_mm ∧Z_mm

!

for p6=m, Xm

k=1

Z_kp∧Z_kq for p6=q, Xm

Zir∧Z_jr for i6=j,

−2m ;

Xm

i,p=1

Zip∧Z_ip.

(4) In this case, ˆRhas four eigenvalues 0,−m,−nand−(m+n), whose eigenvectors are given respectively by

0 ; Z_ip∧Z_ip−Z_in∧Z_in−Z_mp∧Z_mp+Z_mn∧Z_mn for i6=m and p6=n, Z_ir ∧Z_jr−Z_in∧Z_jn for i6=j and r6=n,

Zkp ∧Z_kq−Zmp∧Zmq for k6=m and p6=q, Z_ip∧Z_jq for i6=j and p6=q,

Zα∧Zβ, Zα∧Z_β for any α, β,

−m ;

Xm

k=1

¡Z_kp∧Z_kp−Z_kn∧Z_kp¢

for p6=n Xm

k=1

Zkp∧Z_kq for p6=q,

−n ;

Xn

r=1

(Z_ir∧Z_ir−Z_mr∧Z_mr) for i6=m, Xn

r=1

Zir∧Z_jr for i6=j,

−(m+n) ; X

i,p

Z_ip∧Z_ip.

2 Type II

_m

LetM be an open setD^I_m,n ={ζ ∈M(n;C)|^tζ =−ζ, In−^tζζ >¯ 0}ofC^n(n−1)/2, where Inis the n×nidentity matrix, ^tζ is the transpose ofζ ∈M and ¯ζ denotes the complex conjugate ofζ ∈M. Note that M is a subset of D^I_n,m.

Denote by ζ = (zij) ∈ M with zij = xij +√

−1yij, i, j = 1, . . . , n. Since ζ is a skew-symmetric matrix for all ζ ∈ M, we have zij = −zji and hence the components

z_ij for i < j is the canonical complex coordinate system of M. Let Φ be a real-valued function in a coordinate neighborhoodU at the origin 0 of M defined by

Φ(ζ) = 1

2log det¡

I_n−^tζζ¯ ¢₋₁

=X

i<j

|z_ij|²+1 4

Xz¯_ikz_ilz¯_jlz_jk+ (higher order terms)

for any ζ ∈U.

Now, Greek indicesα, β, . . .denote all subscripts appearing as pairs{12, . . . ,1n,23, . . . ,(n−1)n}, while Latin capitalsA, B, . . .denote{12, . . . ,(n−1)n,12, . . . ,(n−1)n}, and we set

Z_α = ∂

∂zα

= 1 2

µ ∂

∂xα

−√

−1 ∂

∂yα

¶ , Z_α = ∂

∂z¯α

= 1 2

µ ∂

∂xα

+√

−1 ∂

∂yα

¶ .

Then the complexification of the tangent spaceT0M at the origin 0 ofM is represented as

T₀^CM = span_C©

Z_ij, Z_ij |1≤i < j ≤nª . The K¨ahler metric g of M is given, at the origin 0, by

g_αβ = ∂²Φ

∂zα∂zβ

(0), g_αβ = ∂²Φ

∂z_α∂z¯_β(0) ¡

=g_βα¢ , g_{α β} = ∂²Φ

∂z¯_α∂z¯_β(0), whereg_AB =g(Z_A, Z_B)(0). Thus we have

g_{ij kl} =g_{kl ij} =δikδjl, gij kl=g_{ij kl}= 0.

LetR be the curvature tensor of the K¨ahler manifold (M, g), and define R_ABCD = g(R(Z_C, Z_D)Z_B, Z_A)(0). Then we obtain

Rij kl pq rs = ∂⁴Φ

∂z ∂z ∂z ∂z (ζ)

¯¯

¯¯

=δ_pq^rlδ_ij^ks−δ_pq^rkδ_ij^ls−δ_pq^slδ_ij^kr+δ_pq^skδ_ij^lr,

whereδ_ij^kl =∂z_kl/∂z_ij =δ_ikδ_jl−δ_ilδ_jk. By the symmetry properties of R, we also have R_ABCD =R_CDAB =−R_BACD and R_αβCD =R_αβCD = 0.

Then it is immediate to see that the curvature operator ˆR of (M, g) is given by R(Zˆ ij ∧Z_kl) = X

p,q,r,s

¡−δ^rl_pqδ^ks_ij +δ_pq^rkδ_ij^ls+δ_pq^slδ_ij^kr−δ_pq^skδ_ij^lr¢

Zrs∧Zpq, R(Zˆ α∧Zβ) = ˆR(Zα∧Z_β) = 0.

We divide our investigation into the following two cases: (1) n= 2; (2) n≥3.

(1) In this case, ˆR has two eigenvalues 0 and −2, whose eigenvectors are the fol-lowing:

0 ; Z₁₂∧Z₁₂, Z₁₂∧Z₁₂,

−2 ; Z₁₂∧Z₁₂.

(2) In this case, ˆRhas three eigenvalues 0,−(n−2) and−2(n−1), whose eigenvectors are given respectively by

0 ; Z_ij ∧Z_ij −Z_in∧Z_in

−Z_jn∧Z_jn−Z_{(n−2)(n−1)}∧Z_{(n−2)(n−1)}

+Z_(n−2)n∧Z_(n−2)n+Z_(n−1)n∧Z_(n−1)n for 1≤i < j ≤n−2, Zi(n−1)∧Z_i(n−1)−Z(n−1)n∧Z_(n−1)n

−Z_in∧Z_in−Z_{(n−2)(n−1)}∧Z_{(n−2)(n−1)}

+Z_(n−1)n∧Z_(n−1)n+Z_(n−2)n∧Z_(n−2)n for 1≤i≤n−3, Zij ∧Z_ik−Zjn∧Z_kn for 1≤i < j < k ≤n−1,

Z_ij ∧Z_jk+Z_in∧Z_kn for 1≤i < j < k≤n−1, Z_ik∧Z_jk −Z_in∧Z_jn for 1≤i < j < k ≤n−1, Z_ik∧Z_ij −Z_kn∧Z_jn for 1≤i < j < k ≤n−1, Z_jk ∧Z_ij+Z_kn∧Z_in for 1≤i < j < k≤n−1, Z_jk ∧Z_ik−Z_jn∧Z_in for 1≤i < j < k ≤n−1, Z_ij ∧Z_in+Z_j(n−1)∧Z_(n−1)n for 1≤i < j ≤n−2, Z_ij ∧Z_jn−Z_i(n−1)∧Z_(n−1)n for 1≤i < j ≤n−2,

Z_in∧Z_ij +Z_(n−1)n∧Z_j(n−1) for 1≤i < j ≤n−2, Z_jn∧Z_ij −Z_(n−1)n∧Z_i(n−1) for 1≤i < j ≤n−2, Z_i(n−1)∧Z_in−Z_{(n−2)(n−1)}∧Z_(n−2)n for 1≤i≤n−3, Z_in∧Z_i(n−1)−Z_(n−2)n∧Z_{(n−2)(n−1)} for 1≤i≤n−3, Z_ij ∧Z_kl for i6=k, l and j 6=k, l,

Z_α∧Z_β, Z_α∧Z_β for any α, β,

−(n−2) ; − 2 n−2

Xi−1

r=2

Z_ri∧Z_ri+X

r<i

Z_ri∧Z_ri+X

i<r

Z_ir∧Z_ir for 2≤i≤n, Xi−1

r=1

Z_ri∧Z_rj − Xj−1

r=i+1

Z_ir∧Z_rj + Xn

r=j+1

Z_ir∧Z_jr for 1≤i < j ≤n, Xi−1

r=1

Zrj ∧Z_ri− Xj−1

r=i+1

Zrj ∧Z_ir+ Xn

r=j+1

Zjr∧Z_ir for 1≤i < j ≤n,

−2(n−1) ; X

i<j

Z_ij ∧Z_ij.

3 Type III

_m

Let M be an open subset D^III_n = {ζ ∈ M(n;C)|^tζ = ζ, I_n−^tζζ >¯ 0} of C^n(n+1)/2, where I_n is the n×n identity matrix, ^tζ is the transpose of ζ ∈ M and ¯ζ denotes the complex conjugate of ζ ∈M.

Denote by ζ = (z_ij) ∈ M with z_ij = x_ij +√

−1y_ij, i, j = 1, . . . , n. Since ζ is a symmetric matrix for allζ ∈M, we havez_ij =z_ji, so that the componentsz_ij fori≤j is the canonical complex coordinate system of M. Let Φ be a real-valued function in a coordinate neighborhood U at the origin 0 of M defined by

Φ(ζ) = 1

2log det¡

I_n−^tζζ¯ ¢₋₁

= 1 2

Xn

i,j=1

|z_ij|²+ 1 4

Xz¯_ikz_ilz¯_jlz_jk + (higher order terms)

for any ζ ∈U.

In this case, Greek indicesα, β, . . .denote all subscripts appearing as pairs {11,12, . . . , nn}, while Latin capitals A, B, . . .denote {11,12, . . . , mn,11, . . . , nn}, and we set

Z_α = ∂

∂z_α = 1 2

µ ∂

∂x_α −√

−1 ∂

∂y_α

¶ , Z_α = ∂

∂z¯_α = 1 2

µ ∂

∂x_α +√

−1 ∂

∂y_α

¶ .

Then the complexfication of the tangent spaceT₀M at the origin 0 of M is represented as

T₀^CM = span_C©

Z_ij, Z_ji|1≤j ≤j ≤nª . The K¨ahler metric g of M is given, at the origin 0, by

g_{ij kl} = ∂²Φ

∂z_ij∂z_kl(0), g_{ij kl} = ∂²Φ

∂z_ij∂z¯_kl(0) ¡

=g_{kl ij}¢ , g_{ij kl} = ∂²Φ

∂z¯ij∂z¯kl

(0), wheregAB =g(ZA, ZB)(0). Then we have

g_{ij kl} =g_{kl ij} =e^kl_ij for i6=j, k 6=l, g_{ii kk} =g_{ii kk} = 1

2δ_ik, wheree^kl_ij =δ_ikδ_jl+δ_jkδ_il.

LetR be the curvature tensor of the K¨ahler manifold (M, g), and define R_ABCD = g(R(Z_C, Z_D)Z_B, Z_A)(0). Then we obtain

Rij kl pq rs = ∂⁴Φ

∂z_ij∂z_kl∂z_pq∂z_rs(ζ)

¯¯

¯¯

ζ=0

so that

Rii kk pp rr =δ_rpδ_riδ_pkδ_ki,

Rij kl pp rr =δ_rpδ_lpe^kr_ij +δ_rpδ_kpe^lr_ij, Rij kk pq rr =e^kr_ije^kr_pq,

Rij kk pp rs =δ_ikδ_pke^pj_rs+δ_jkδ_pke^pi_rs, Rii kl pp rs=e^pi_rse^pi_kl,

Rii kl pq rs =e^ip_kle^iq_rs+e^iq_kle^ip_rs, Rij kk pq rs=e^kr_pqe^ks_ij +e^ks_pqe^kr_ij,

Rij kl pq rs=e^kr_pqe^ls_ij +e^ks_pqe^lr_ij +e^lr_pqe^ks_ij +e^ls_pqe^kr_ij,

where i < j, k < l, p < q and r < s. By the symmetry properties of R, it is easy to see thatR_ABCD =R_CDAB =−R_BACD and R_αβCD =R_αβCD = 0.

It then is immediate to see that the curvature operator ˆR of (M, g) is given by R(Zˆ _ii∧Z_kk) =−4X

r,p

δ_rpδ_riδ_kpδ_ikZ_rr∧Z_pp− X

p,q,r,s

¡δ_ikδ_ise^kr_pq +δ_ikδ_ire^sk_pq¢

Z_rs∧Z_pq, R(Zˆ ij ∧Z_kk) =−2X

r,p,q

e^kr_ije^kr_pqZrr∧Zpq−2X

p,r,s

¡δikδpke^pj_rs+δjkδpke^pi_rs¢

Zrs∧Zpp

− X

p,q,r,s

¡e^kr_pqe^ks_ij +e^ks_pqe^kr_ij¢

Z_rs∧Z_pq, R(Zˆ _ii∧Z_kl) =−2X

p,q,r

¡δ_prδ_ire^iq_kl+δ_qrδ_ire^ip_kl¢

Z_rr∧Z_pq −2X

p,r,s

e^pi_rse^pi_klZ_rs∧Z_pp

− X

p,q,r,s

¡e^ip_kle^iq_rs+e^iq_kle^ip_rs¢

Z_rs∧Z_pq, R(Zˆ _ij ∧Z_kl) =−4X

p,r

¡δ_rpδ_lpe^kr_ij +δ_rpδ_kpe^lr_ij¢

Z_rr∧Z_pp

−2X

p,q,r

¡e^rk_ije^rl_pq+e^rl_ije^rk_pq¢

Z_rr∧Z_pq

−2X

p,r,s

¡e^pi_rse^pj_kl +e^pj_rse^pi_kl¢

Z_rs∧Z_pp

− X

p,q,r,s

¡e^kr_pqe^ls_ij +e^ks_pqe^lr_ij +e^lr_pqe^ks_ij +e^ls_pqe^kr_ij¢

Zrs∧Zpq.

Consequently, the curvature operator ˆRhas three eigenvalues 0,−(n+2) and−2(n+

1), whose eigenvectors are given respectively by

0 ; Zij ∧Z_ij−Zin∧Z_in−Zjn∧Z_jn+ 2Znn∧Znn for 1≤i < j ≤n−1, Z_ii∧Z_ii−Z_in∧Z_in+Z_nn ∧Z_nn for 1 ≤i≤n−1,

Zij ∧Z_ik−Zjn∧Z_kn for 1≤i≤j < k≤n, Z ∧Z −Z ∧Z for 1≤i < j ≤k≤n−1,

Z_ij ∧Z_jn−Z_in∧Z_jn for 1 ≤i < j≤n−1, Z_ik∧Z_jk−Z_in∧Z_jn for 1≤i < j < k≤n−1, Z_ik∧Z_ij−Z_kn∧Z_jn for 1≤i≤j < k≤n, Z_jk∧Z_ij −Z_kn∧Z_in for 1≤i < j ≤k≤n−1, Z_jn∧Z_ij −Z_nn∧Z_in for 1≤i < j ≤n−1, Z_jk∧Z_ik−Z_jn∧Z_in for 1≤i < j < k≤n−1, Z_ii∧Z_kk for i6=k,

Z_ij ∧Z_kk for i < j and i, j 6=k, Z_ii∧Z_kl for k < l and i6=k, l,

Z_ij ∧Z_kl for i < j, k < l and i6=k, l, j 6=k, l, Z_α∧Z_β, Z_α∧Z_β for any α, β,

−(n+ 2) ; Zii∧Z_ii+ 1 4

Xn

r=i+1

Zir∧Z_ir+ 1 4

Xi−1

r=1

Zri∧Z_ri

−Z_nn∧Z_nn− 1 4

Xn−1

r=1

Z_rn∧Z_rn for 1≤i≤n−1, 1

2 Xi−1

r=1

Z_ri∧Z_rk+Z_ii∧Z_ik+1 2

Xk−1

r=i+1

Z_ir∧Z_rk+Z_ik∧Z_kk + 1

2 Xn

r=k+1

Z_ir∧Z_kr for 1≤i < k≤n, 1

2 Xk−1

p=1

Z_pk∧Z_pi+Z_kk∧Z_ik+1 2

Xi−1

p=k+1

Z_kp∧Z_pi+Z_ki∧Z_ii

+ 1 2

Xn

p=i+1

Zkp∧Z_ip for 1≤i < k ≤n,

−2(n+ 1) ; 2 Xn

r=1

Z_rr∧Z_rr+X

r<s

Z_rs∧Z_rs.

4 Type IV

_m

LetM be an open subsetD_n^IV ={ζ ∈M(n,1);C)|1+|^tζζ|²−2^tζζ >0,^tζζ <¯ 1}ofCⁿ, where^tζ is the transpose ofζ ∈M, and ¯ζ denotes the complex conjugate of ζ ∈M.

Let ζ =^t(z₁, . . . , z_n) with z_i =x_i+√

−1y_i, i = 1, . . . , n be the canonical complex coordinate system ofM. Let Φ be a real-valued function in a coordinate neighborhood U at the origin 0 of M defined by

Φ(ζ) = log det¡

1−2^tζζ+|ζ|²¢₋₁

= 2 Xn

α=1

|z_α|²+X

α,β

(z_α)²(¯z_β)² + 2X

α,β

|z_α|²|z_β|²+ (higher order terms) for any ζ ∈U.

Now, Greek indices α, β, . . . run from 1 to n, while Latin capitals A, B, . . . run through 1, . . . , n,¯1, . . . ,n, and we set¯

Z_α = ∂

∂zα

= 1 2

µ ∂

∂xα

−√

−1 ∂

∂yα

¶ , Z_α = ∂

∂z¯α

= 1 2

µ ∂

∂xα

+√

−1 ∂

∂yα

¶ . The K¨ahler metric g of M is given, at the origin 0, by

g_αβ = ∂²Φ

∂zα∂zβ

(0), g_αβ = ∂²Φ

∂z_α∂z¯_β(0), ¡

=g_βα¢ g_αβ = ∂²Φ

∂z¯_α∂z¯_β(0), whereg_AB =g(Z_A, Z_B)(0). Then we have

g_αβ(0) =g_βα(0) =δijδαβ.

LetR be the curvature tensor of the K¨ahler manifold (M, g), and define R_ABCD = g(R(Z_C, Z_D)Z_B, Z_A)(0). Then we have

R_αβγδ(0) = ∂⁴Φ

∂z_α∂z_β∂z_γ∂z_δ(ζ)

¯¯

¯¯

ζ=0

=−4δ δ + 4δ δ + 4δ δ ,

and it is easy to see thatR_ABCD =R_CDAB =−R_BACD and R_αβCD =R_αβCD = 0.

Then it is immediate to see that the curvature operator ˆR of (M, g) is given by R(Zˆ _α∧Z_α) = −

Xn

γ=1

Z_γ∧Z_γ,

R(Zˆ _α∧Z_β) = Z_β∧Z_α−Z_α∧Z_β, for α6=β.

Then ˆR has three eigenvalues 0,−2 and −n, whose eigenvectors are given respectively by

0 ; Z_α∧Z_α−Z_n∧Z_n, for 1≤α≤n−1, Zα∧Z_β +Zβ∧Zα for α6=β,

−2 ; Zα∧Z_β −Zβ∧Zα, for α6=β,

−n ;

Xn

α=1

Z_α∧Z_α.

ドキュメント内 Homogeneous Kähler Einstein manifolds of nonpositive curvature operator (ページ 119-131)