Topological property of the holonomy displacement on the principal U(n) -bundle over D n,m , related to complex surfaces
aa r X i v : . [ m a t h . DG ] S e p TOPOLOGICAL PROPERTY OF THE HOLONOMYDISPLACEMENT ON THE PRINCIPAL U ( n ) -BUNDLEOVER D n,m , RELATED TO COMPLEX SURFACES
TAECHANG BYUN
Abstract.
Consider D n,m = U ( n, m ) / ( U ( n ) × U ( m )), the dual of thethe Grassmannian manifold and the principal U ( n ) bundle over D n,m ,U ( n ) → U ( n, m ) /U ( m ) π → D n,m . Given a nontrivial X ∈ M m × n ( C ) , consider a two dimensional subspace m ′ ⊂ m ⊂ u ( n, m ) , induced by X, iX ∈ M m × n ( C ) , and a complete oriented surface S, related to( X, g ) ∈ M m × n ( C ) × U ( n, m ) , in the base space D n,m with a com-plex structure from m ′ . Let c be a smooth, simple, closed, orientation-preserving curve on S parametrized by 0 ≤ t ≤
1, and ˆ c its horizon-tal lift on the bundle U ( n ) → U ( n, m ) /U ( m ) π → D n,m . Then theholonomy displacement is given by the right action of e Ψ for someΨ ∈ Span R { i ( X ∗ X ) k } qk =1 ⊂ u ( n ) , q = rk X, such thatˆ c (1) = ˆ c (0) · e Ψ and Tr(Ψ) = 2 i Area( c ) , where Area( c ) is the area of the region on the surface S surrounded by c, obtained from a special 2-form ω ( X,g ) on S, called an area form ω ( X,g ) related to ( X, g ) on S. Introduction
Gauss-Bonnet Theorem shows a kind of relation between Riemannian Ge-ometry and Topology through two kinds of curvatures -Gaussian curvatureand geodesic curvature-(, angles if needed) and Euler-characterstic. In thispaper, we explain a similar phenomenon in some principal bundles througharea and holonomy displacement.In [6], Pinkall showed that the holonomy displacement of a simple closedcurve in the base space on the Hopf bundle S → S → S depends on thearea of its interior. Byun and Choi [1] generalized this result to the principal U ( n )-bundle over the Grassmannian manifold G n,m of complex n -planes in C n + m , U ( n ) → U ( n + m ) /U ( m ) → G n,m , by introducing U m,n ( C ) , where U m,n ( C ) := { X ∈ M m × n ( C ) | X ∗ X = λI n for some λ ∈ C − { }} . Mathematics Subject Classification.
Key words and phrases.
Holonomy displacement, Area form, Riemannian submersion,complex surface, principal bundle, Lie algebra, complete totally geodesic submanifold,Hermition form, Grassmannian manifold.
Especially, the result related to a complex surface in G n,m can be summa-rized as follows: Theorem 1.1. [1]
Assume U ( k ) , k = 1 , , · · · , has a metric, related to theKilling-Cartan form, given by h A, B i = k Re (cid:0) Tr ( A ∗ B ) (cid:1) , A, B ∈ u ( k ) , Consider a bundle U ( n ) → U ( n + m ) /U ( m ) pr −→ G n,m , where pr : U ( n + m ) /U ( m ) → G n,m is a Riemannian submersion. Given a nontriv-ial X ∈ U m,n ( C ) , a two dimensional subspace m ′ ∈ u ( n + m ) , induced by X and iX, gives rise to a complete totally geodesic surface S with a complexstructure in the base space G n,m . And if c : [0 , → S is a piecewise smooth,simple, closed curve on S and if ˜ c is its horizontal lift, then the holonomydisplacement along c , e c (1) = e c (0) · V ( c ) , is given by the right action of V ( c ) = e iθ I n ∈ U ( n ) , where A ( c ) is the inducedarea of the region, surrounded by c, on the surface S, from the metric on G n,m , and θ = 2 · n + m n A ( c ) . If the metric is changed into h A, B i = Re (cid:0) Tr( A ∗ B ) (cid:1) , A, B ∈ u ( k ) , then the the result will be done into θ = 2 · n A ( c ) , and from e iθ I n = e iθI n , it can be read as iθI n ∈ u ( n ) and Tr( iθI n ) = inθ = 2 i A ( c ) , where A ( c ) is the changed area induced from the changed metric on G n,m . On the other hand, Choi and Lee [2] thought of the dual version ofPinkall’s result over C H m : Theorem 1.2. [2]
Let S → S m, → C H m be the natural fibration. Let S be a complete totally geodesic surface in C H m , and ξ S be the pullbackbundle over S. Let c be a piecewise smooth, simple closed curve on S. Thenthe holonomy displacement along c is given by V ( c ) = e A ( c ) i or e i ∈ S , where A ( c ) is the area of the region on the surface S surrounded by c, depending on whether S is a complex submanifold or not. In [2], the bundle S → S m, → C H m is being studied through the one U (1) → U (1 , m ) /U ( m ) → U (1 , m ) / ( U (1) × U ( m )) with U (1 , m ) /U ( m ) ∼ = S m, and U (1 , m ) / ( U (1) × U ( m )) ∼ = C H m , and the unmentioned diffeomorphism f : C H m −→ U (1 , m ) / ( U (1) × U ( m )) is a conformal map satisfying | f ∗ v | = | v | . HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m Consider D n,m = U ( n, m ) / (cid:0) U ( n ) × U ( m ) (cid:1) . We generalize these results in the principal U ( n )-bundle U ( n ) → U ( n, m ) /U ( m ) π → D n,m over D n,m up to M m × n ( C ) for general positive integers n, m ∈ N , not only upto U m,n ( C ) , as follows: consider a left invariant metric on U ( n, m ), relatedto the Killing-Cartan form, given by h A, B i = Re (cid:0) Tr( A ∗ B ) (cid:1) , A, B ∈ u ( n, m ) , (1–1)and the induced metric on D n,m , which makes the natural projection˜ π : U ( n, m ) −→ D n,m a Riemannian submersion and induces another Riemannain submersion π : U ( n, m ) /U ( m ) −→ D n,m . Given a nontrivial X ∈ M m × n ( C ) , let b X = (cid:18) O n X ∗ X O m (cid:19) and c iX = (cid:18) O n − iX ∗ iX O m (cid:19) . Then for W = | b X | X, for a, b ∈ R and for z = a + bi ∈ C d zW = \ a + bi (cid:12)(cid:12) b X (cid:12)(cid:12) X = a (cid:12)(cid:12) b X (cid:12)(cid:12) b X + b (cid:12)(cid:12) c iX (cid:12)(cid:12) c iX = a c W + b c iW . Consider a complete surface ˜ S = { e d zW | z ∈ C } in U ( n, m ) . Then the map z e d zW : C → ˜ S is a bijection. For g ∈ U ( n, m ) , define a complex surface S related to ( X, g ) in D n,m by S = ˜ π ( g ˜ S ) , which has a complex structureinduced from a 2-dimensional subspace m ′ = Span R { b X, c iX } = { d zW | z = x + iy for x, y ∈ R } ⊂ m ⊂ u ( n, m ) , where m is the orthogonal complementof u ( n ) + u ( m ) . In case of U (1) → U (1 , m ) /U ( m ) → U (1 , m ) / ( U (1) × U ( m ))and in case of U ( n ) → U ( n, /U (1) → U ( n, / ( U ( n ) × U (1)) , the Lie algebra generated by { b X, c iX } from X ∈ M m × n ( C ) , either m = 1or n = 1 , inducing a complex surface S in the base space, produces a 3-dimensional Lie subgroup b G of U ( n, m ) , which is isomorphic to SU (1 , , and a bundle structure isomorphic to S ( U (1) × U (1)) −→ SU (1 , −→ SU (1 , /S ( U (1) × U (1)) . TAECHANG BYUN
Then, under the notaion A ( c ) of the induced area of the region surroundedby c on the surface S from the metric on the base space, this enables us toguess the pull-back bundle and the holonomy displacement V ( c ) = e A ( c ) i along a curve c in the base space [2], which also enables us to guess itsinduced holonomy displacement in the original bundle from either X ∗ X ∈ R or XX ∗ ∈ R , from Lemma 2.2 and from Proposition 1.4. We deal with thelatter one in Section 3.But for general positive integers n, m ∈ N and for a general nontrivial X ∈ M m × n ( C ) , the Lie algebra generated by { b X, c iX } is not 3-dimensional.Furthermore, the holonomy displacement depends not only on X but alsoon some 2-form of the complex surface S related to it too heavily. Fromnow on, we consider two kinds of 2-forms on S, the first one is related to( X, g ) ∈ M m × n ( C ) × U ( n, m ) , defined in Definition 1.3, and the other one isinduced from the metric on S obtained from the metric on D n,m , mentionedin Proposition 1.4.To begin with, given X ∈ M m × n ( C ) , think of W = | b X | X and ˜ S = { e d zW | z ∈ C } , which is one to one correspondent to S := ˜ π ( ˜ S ) . Refer toRemark 1.11. And for any ˜ g ∈ U ( n, m ) , let L ˜ g : D n,m → D n,m be the ac-tion of ˜ g, induced from the left multiplication of ˜ g on U ( n, m ) , which is anisometry from L ˜ g ◦ ˜ π = ˜ π ◦ L ˜ g . Definition 1.3.
Think of a bundle U ( n ) × U ( m ) → U ( n, m ) ˜ π −→ D n,m . Given ( X, g ) ∈ M m × n ( C ) × U ( n, m ) , consider W = | b X | X and define a 2-form ω ( X,g ) on a complex surface S related to ( X, g ) in D n,m , called an areaform ω ( X,g ) related to ( X, g ) on S, by ω ( X,g ) (˜ π ∗ x, ˜ π ∗ y ) = det h L g − ∗ x, c W i h L g − ∗ y, c W ih L g − ∗ x, c iW i h L g − ∗ y, c iW i ! under the identification of the tangent space of U ( n, m ) at the identity andits Lie algebra u ( n, m ) , where g ∈ g ˜ S = { ge z c W | z ∈ C } and both x and y are tangent to g ˜ S at g . Proposition 1.4.
Think of a bundle U ( n ) × U ( m ) → U ( n, m ) ˜ π −→ D n,m such that ˜ π is a Riemannian submersion. Given a nontrivial X ∈ M m × n ( C ) , consider a complete, complex surface ˜ S = { e d zW (cid:12)(cid:12) z ∈ C } in U ( n, m ) for W = | b X | X and for c W = \ (cid:12)(cid:12) b X (cid:12)(cid:12) X = (cid:12)(cid:12) b X (cid:12)(cid:12) b X. (i) For a complex surface S = ˜ π ( ˜ S ) related to ( X, e ) in D n,m , let ω As explained in the footnote to A ( c ) in Theorem 1.2, V ( c ) in [2] is given by e A ( c ) i . HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m be the area form ω ( X,e ) related to ( X, e ) on S , where e is the identity of U ( n, m ) . Consider a coordinate system ( r, θ ) on S induced from S = { e \ re iθ W (cid:12)(cid:12) z = re iθ ∈ C } . Then, for the differntial d = d S on S ,ω = ω ( X,e ) = d (cid:16) n X j =1 12 sinh ( σ j r ) dθ (cid:17) , where σ ≥ · · · ≥ σ n are the square roots of decreasingly ordered non-negativeeigenvalues of W ∗ W with σ + · · · + σ n = 1 . (Refer to Lemma 2.2 for moreinformation on the eigenvalues.) (ii)
Given a complex surface S in D n,m , related to ( X, g ) ∈ M m × n ( C ) × U ( n, m ) , the area form ω ( X,g ) related to ( X, g ) on S is given by ω ( X,g ) = L g − ∗ ω . (iii) The induced area form ω on S from the metric on D n,m is given by ω = L ∗ g − ω and ω = 12 vuut n X j =1 sinh (2 σ j r ) dr ∧ dθ, where ω is the induced area form on S from the metric on D n,m . Especially, if the positive eigenvalues of W ∗ W consist of a single realnumber with allowing the duplication, that is, σ = · · · = σ q > σ q +1 = · · · = σ n , then ω = ω = d (cid:0) q sinh (cid:0) r √ q (cid:1) dθ (cid:1) and q ∈ N is the algebraic multiplicity of the positive eigenvalue σ of W ∗ W. Theorem 1.5.
Consider a bundle U ( n ) → U ( n, m ) /U ( m ) π → D n,m suchthat π is a Riemannian submersion. Given a complex surface S in D n,m , related to ( X, g ) ∈ M m × n ( C ) × U ( n, m ) , let c be a smooth, simple, closed,oreintation-preserving curve on S, parametrized by ≤ t ≤ , and ˆ c itshorizontal lift. Then the holonomy displacement ˆ c (1) = ˆ c (0) · e Ψ is given bythe right action of e Ψ for some Ψ ∈ Span R { i ( X ∗ X ) k } qk =1 ⊂ u ( n ) , q = rk X, such that Tr(Ψ) = 2 i Area( c ) , where rk X is the rank of X and Area( c ) is the area of the region on thesurface S surrounded by c with respect to the area form ω ( X,g ) related to ( X, g ) on S. (For more information on how to find a concrete Ψ ∈ u ( n ) not only for n = 1 but also for n > , refer to Remark
TAECHANG BYUN
Remark 1.6.
One metric structure on D n,m induced from a Riemanniansubmersion ˜ π : U ( n, m ) → D n,m and the other one induced from a Riemann-ian submersion π : U ( n, m ) /U ( m ) → D n,m , in fact, are same. See Section2 . Corollary 1.7.
In addition to the hypothesis of Theorem 1.5, assume that X ∗ X has a single positive eigenvalue with algebraic multiplicity q. Then,
Ψ = A diag[ iq Area( c ) , · · · , iq Area( c ) , | {z } q times , · · · , | {z } ( n − q ) times ] A ∗ = A diag[ iq A ( c ) , · · · , iq A ( c ) , | {z } q times , · · · , | {z } ( n − q ) times ] A ∗ for some A ∈ U ( n ) , where A ( c ) is the area with respect to the induced metricon S from the metric on D n,m , i.e., the area with respect to the induced areaform ω on S from the metric on D n,m . Especially, if q = n, then X ∈ U m,n ( C ) , S is totally geodesic and e Ψ = e iθ I n with θ = n Area( c ) = n A ( c ) Proof.
From hypothesis, for W = | b X | X, W ∗ W also has a single positiveeigenvalue with algebraic multiplicity q. Call its positive square root σ. Then,under the notation of Lemma 2.2, we get, for some A ∈ U ( n ) ,W ∗ W = A Σ ∗ Σ A ∗ = A diag[ σ , · · · , σ , | {z } q times , · · · , | {z } ( n − q ) times ] A ∗ = A diag[ q , · · · , q , | {z } q times , · · · , | {z } ( n − q ) times ] A ∗ from Proposition 1.4. Then, Theorem 1.5 and Proposition 1.4 say thatΨ ∈ Span { X ∗ X } = Span { W ∗ W } with Tr(Ψ) = 2 i Area( c ) = 2 iA ( c ) , which proves the former part of Corollary.For the latter one, assume q = n. Then the similar arguments as thosefor W ∗ W = A Σ ∗ Σ A ∗ before show that X ∗ X = ˜ A ( λI n ) ˜ A = λI n ∈ U m,n ( C )(1–2)for some λ > A ∈ U ( n ) . And q = n impliesΨ = A (cid:16) in Area( c ) I n (cid:17) A ∗ = in Area( c ) I n = iθI n for θ = n Area( c ) and for some A ∈ U ( n ) , which gives e Ψ = e iθI n = e iθ I n . HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m Furthermore, note that W = √ nλ X and W ∗ W = n I n from equation (1–2). Then for b V = (cid:18) iW ∗ W OO − iW W ∗ (cid:19) = (cid:18) in I n OO − iW W ∗ (cid:19) , a set { b X, c iX, b V } , or { √ λ b X, √ λ c iX, n b V } = {√ n c W , √ n c iW , n b V } generates a 3-dimensional Lie algebra with[ √ n c W , √ n c iW ] = 2 n b V , [ n b V , √ n c W ] = − √ n c iW and [ n b V , √ n c iW ] = 2 √ n c W .
Since each action of g ∈ U ( n, m ), L g : D n,m → D n,m is an isometry from L g ◦ ˜ π = ˜ π ◦ L g and S = L g ( S ) , Proposition 2.3 says that S is totallygeodesic. (cid:3) Remark 1.8.
For general q = 1 , · · · , n, we can show that S in Corollary 1.7 is totally geodesic by using singular value decomposition: under the notationof
Lemma 2.2 , we get, for some B ∈ U ( m ) ,W W ∗ = B ΣΣ ∗ B ∗ = B diag[ σ , · · · , σ , | {z } q times , · · · , | {z } ( m − q ) times ] B ∗ = B diag[ q , · · · , q , | {z } q times , · · · , | {z } ( m − q ) times ] B ∗ . If we let
Ω = (cid:18)
A OO B (cid:19) and b V = (cid:18) iW ∗ W OO − iW W ∗ (cid:19) , then by using singular value decomposition, we get a set { b X, c iX, b V } , or {√ q c W , √ q c iW , q b V } generates a 3-dimensional Lie algebra with [ √ q c W , √ q c iW ] = 2 q b V , [ q b V , √ q c W ] = − √ q c iW and [ q b V , √ q c iW ] = 2 √ q c W , so Proposition 2.3 says that S is totally geodesic. TAECHANG BYUN
Remark 1.9. Ψ in Theorem 1.5 is obtained by a solution of a system offirst order linear differential equations: for t ∈ [0 , , consider a curve z ( t ) = r ( t ) e iθ ( t ) in C and another one ˜ γ ( t ) in ˜ S ⊂ U ( n, m ) such that ˜ γ ( t ) = e \ z ( t ) W and that ˜ π ◦ L g ◦ ˜ γ = c, where W = (cid:12)(cid:12) b X (cid:12)(cid:12) X and L g : U ( n, m ) → U ( n, m ) is anisometry by the left multiplication of g ∈ U ( n, m ) . For a curve Ψ( t ) ∈ u ( n ) , given by Ψ( t ) = (cid:18) i P qk =1 φ k ( t )( W ∗ W ) k OO O m (cid:19) ∈ u ( n ) , let ˆ c ( t ) = (cid:0) g ˜ γ ( t ) e Ψ( t ) (cid:1) U ( m ) be a horizontal lift of c ( t ) , where φ k ( t ) is tobe determined for each k = 1 , · · · q. Then Ψ is given by Ψ = Ψ(1) − Ψ(0) . Furthermore, for some A ∈ U ( n ) and B ∈ U ( m ) from Lemma 2.2 , we willget Ψ( t ) = Ω diag h i q X k =1 σ k φ k ( t ) , · · · , i q X k =1 σ kq φ k ( t ) , , · · · , | {z } ( n − q ) times , , · · · , | {z } m times i Ω ∗ , where Ω = (cid:18)
A OO B (cid:19) , and a system of a first order linear differential equations q X k =1 σ kj φ ′ k ( t ) = θ ′ ( t ) sinh ( σ j r ( t )) , j = 1 , · · · , q. (1–3) Refer to
Section 5 to see why this system comes. And for the concreteexpression of Ψ( t ) , refer to Remark
Remark 1.10.
For simplicity, we can regard Ψ( t ) in Remark 1.9 as Ψ( t ) = i q X k =1 φ k ( t )( W ∗ W ) k ∈ u ( n ) . Then we get Ψ( t ) = A diag h i q X k =1 σ k φ k ( t ) , · · · , i q X k =1 σ kq φ k ( t ) , , · · · , | {z } ( n − q ) times i A ∗ . If { σ , · · · , σ q } = { σ j , · · · , σ j p } with σ j > · · · > σ j p , thenfor uniquely determined ( n × n ) -matrices I j l ’s from the equation HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m diag[ σ , · · · , σ q , , · · · ,
0] = P pl =1 σ j l I j l , we have W ∗ W = A diag[ σ , · · · , σ q , , · · · , | {z } ( n − q ) times ] A ∗ = A (cid:0) p X l =1 σ j l I j l (cid:1) A ∗ = p X l =1 σ j l AI j l A ∗ . Note ( W ∗ W ) k = A diag[ σ k , · · · , σ kq , , · · · , | {z } ( n − q ) times ] A ∗ = A (cid:0) p X l =1 σ kj l I j l (cid:1) A ∗ = p X l =1 σ kj l AI j l A ∗ for k ∈ { , · · · , q } , which says that Span R { i ( X ∗ X ) k } qk =1 in Theorem 1.5 hasa dimension p and a basis { AI j l A ∗ } pl =1 . For k = 1 , · · · , p, this system canbe regarded as σ · j · · · σ · j p ... . . . ... σ pj · · · σ pj p AI j A ∗ ... AI j p A ∗ = ( W ∗ W ) ... ( W ∗ W ) p . Then for the invertible ( p × p ) -matrix D on the left hand side, D = σ · j · · · σ · j p ... . . . ... σ pj · · · σ pj p , we get AI j A ∗ ... AI j p A ∗ = D − ( W ∗ W ) ... ( W ∗ W ) p , which says that { ( W ∗ W ) k } pk =1 is also a basis. From this basis, the curve Ψ( t ) may be reconstructed by Ψ( t ) = p X k =1 iψ k ( t )( W ∗ W ) k = A (cid:16) p X l =1 (cid:0) p X k =1 iψ k ( t ) σ kj l (cid:1) I j l (cid:17) A ∗ , which change a system of first order differential equations (1–3) for con-structing a horizontal curve condition into another one p X k =1 σ kj l ψ ′ k ( t ) = θ ′ ( t ) sinh ( σ j l r ( t )) , l = 1 , · · · , p, whose intial value is given by Ψ(0) satisfying ˆ c (0) = (cid:0) g ˜ γ (0) e Ψ(0) (cid:1) U ( m ) . This system can be rewritten as σ · j · · · σ pj ... . . . ... σ · j p · · · σ pj p ψ ′ ( t ) ... ψ ′ p ( t ) = θ ′ ( t ) sinh ( σ j r ( t )) ... θ ′ ( t ) sinh ( σ j p r ( t )) . Then for the invertible ( p × p ) -matrix C on the left hand side, C = σ · j · · · σ pj ... . . . ... σ · j p · · · σ pj p = D T , we get a solution ψ ( t ) ... ψ p ( t ) = Z C − θ ′ ( t ) sinh ( σ r ( t )) ... θ ′ ( t ) sinh ( σ p r ( t )) dt. Especially, if X ∈ U m,n ( C ) , then p = 1 , σ = · · · = σ n , Span R { i ( X ∗ X ) k } qk =1 = { iµI n × n | µ ∈ R } , and ψ ( t ) = Z σ θ ′ ( t ) sinh ( σ r ( t )) dt, so Ψ( t ) is given by Ψ( t ) = A ( iσ ψ ( t ) I n × n ) A ∗ = iσ ψ ( t ) I n × n = Ψ(0) + i (cid:16) Z t θ ′ ( u ) sinh ( σ r ( u )) du (cid:17) I n × n . HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m Remark 1.11.
The singular value decomposition of e d zW , consisting of Ω , Γ n ( z ) , Γ m ( z ) and Λ( z ) , in Section 4 shows that z e d zW : C → ˜ S isa bijection. Furthermore, it also shows that the restriction of ˜ π on ˜ S, ˜ π : ˜ S → S is injective(, so bijective): to show it, assume that ˜ π ( e d z W ) = ˜ π ( e d z W ) for z j = r j e iθ j , j = 1 , , with r j ≥ . Then, we get e \ − z W e d z W ∈ U ( n ) × U ( m ) and the calculation through their singular value decomposition gives e \ − z W e d z W = Ω (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) − (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) Ω ∗ = Ω (cid:18) Γ n ( z ) − Λ( z ) ∗ − Λ( z ) Γ m ( z ) (cid:19) (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) Ω ∗ , and so, from Ω ∈ U ( n ) × U ( m ) , (cid:18) Γ n ( z ) − Λ( z ) ∗ − Λ( z ) Γ m ( z ) (cid:19) (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) ∈ U ( n ) × U ( m ) , whose (1 , n + 1) -element is (cid:16) cosh( σ r ) 0 · · · | {z } ( n −
1) times − e − iθ sinh( σ r ) 0 · · · | {z } ( m −
1) times (cid:17) e − iθ sinh( σ r )0 ... σ r )0 ... = e − iθ cosh( σ r ) sinh( σ r ) − e − iθ cosh( σ r ) sinh( σ r )= (cid:0) cos θ cosh( σ r ) sinh( σ r ) − cos θ cosh( σ r ) sinh( σ r ) (cid:1) − i (cid:0) sin θ cosh( σ r ) sinh( σ r ) − sin θ cosh( σ r ) sinh( σ r ) (cid:1) . Then cos θ cosh( σ r ) sinh( σ r ) = cos θ cosh( σ r ) sinh( σ r ) , (1–4) sin θ cosh( σ r ) sinh( σ r ) = sin θ cosh( σ r ) sinh( σ r ) , (1–5) so from cos θ + sin θ = 1 , cosh ( σ r ) sinh ( σ r ) = cosh ( σ r ) sinh ( σ r ) , which means either σ r ) sinh( σ r ) − cosh( σ r ) sinh( σ r )= sinh( σ ( r − r )) or σ r ) sinh( σ r ) + cosh( σ r ) sinh( σ r )= sinh( σ ( r + r )) . Thus we get r = r ≥ . If both of them equals to , then z = z = 0 . If r = r > , then the equations (1–4) and (1–5) say that cos θ = cos θ and sin θ = sin θ , which gives z = r e iθ = r e iθ = z . Preliminaries
Given a submersion pr : ( M, g M ) → ( B, g B ) , the vertical distribution V and the horizontal distribution H = V ⊥ are defined to be the kernelof pr ∗ and its orthogonal complement, respectively. And pr is said to be Riemannian if | pr ∗ x | = | x | for all x ∈ H . [3]Let K be a subgroup of the isometry group of a Riemannian manifold M, and suppose that all orbits have the same type, that is, any two areequivaiantly diffeormorphic. Then there exits a differentiable structure on M/K with a Riemannian metric for each the natural projection π : M → M/K is a Riemannian submersion. [3]Given a Lie group G with a left-invariant metric and given a subgroup H with the right multiplication R h an isometry for each h ∈ H, the spaceof left cosets G/H can be endowed with the metric for which the canonicalprojection π : G → G/H is a Riemannain submersion. If we denote by L g : G/H → G/H the action of g ∈ G, then L g ◦ π = π ◦ L g and so L g is anisometry of G/H. [3]Recall that G = U ( n, m ) = { Φ ∈ GL n + m ( C ) | Φ ∗ Λ nm Φ = Λ nm } = { Φ ∈ GL n + m ( C ) (cid:12)(cid:12) F (Φ v, Φ w ) = F ( v, w ) , v, w ∈ C n + m } and that D n,m := U ( n, m ) / ( U ( n ) × U ( m )) can be regarded as the set of n -dimensional subspaces V of C n + m such that F ( v, v ) ≤ v ∈ V [5], where F : C n + m → C is an Hermitian form defined by F ( v, w ) = v ∗ Λ nm w = − n X k =1 ¯ v k w k + n + m X s = n +1 ¯ v s w s for column vectors v, w ∈ C n + m , Λ nm = (cid:18) − I n O n × m O m × n I m (cid:19) . HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m Consider the following canonical decomposition of the Lie algebra u ( n, m )of G = U ( n, m ): u ( n, m ) = h + m , where h = u ( n ) + u ( m ) = (cid:26)(cid:18) A O n × m O m × n B (cid:19) : A ∈ u ( n ) , B ∈ u ( m ) (cid:27) and m = (cid:26) ˆ X := (cid:18) O n X ∗ X O m (cid:19) : X ∈ M m × n ( C ) (cid:27) . Since the right multiplication R h , h ∈ U ( n ) × U ( m ) , is an isometry withrespect the left invariant metric given by the equaion (1–1), there are twokinds of principal bundles U ( m ) → U ( n, m ) ˆ π −→ U ( n, m ) /U ( m )and U ( n ) × U ( m ) → U ( n, m ) ˜ π −→ D n,m such that both ˆ π and ˜ π are Riemannian submersions. Note that each actionof g ∈ U ( n, m ) on U ( n, m ) /U ( m ) and on D n,m , denoted by b L g : U ( n, m ) /U ( m ) −→ U ( n, m ) /U ( m ) , and L g : D n,m −→ D n,m , respectively, is an isometry from b L g ◦ ˆ π = ˆ π ◦ L g and L g ◦ ˜ π = ˜ π ◦ L g . For each h ∈ U ( n ) , consider the right action b R h : U ( n, m ) /U ( m ) → U ( n, m ) /U ( m )given by b R h (cid:0) gU ( m ) (cid:1) = (cid:0) gU ( m ) (cid:1) · h := ( gh ) U ( m ) , where h = (cid:18) h O n × m O m × n I m (cid:19) . This action is well-defined from (cid:18) h OO I m (cid:19) (cid:18) I n OO h (cid:19) = (cid:18) I n OO h (cid:19) (cid:18) h OO I m (cid:19) . for any h ∈ U ( m ) . By abusing of notations, write R h = R h . Then the equa-tion b R h ◦ ˆ π = ˆ π ◦ R h implies that b R h : U ( n, m ) /U ( m ) → U ( n, m ) /U ( m )is an isometry since R h : U ( n, m ) → U ( n, m ) is an isometry preservingeach fiber of the bundle ˆ π : U ( n, m ) → U ( n, m ) /U ( m ) . More concretely,for any horizontal vector x with respect to the Riemannian submersionˆ π : U ( n, m ) → U ( n, m ) /U ( m ) , R h ∗ x is a horizontal vector since R h is an isometry preserving the fibers of the bundle ˆ π : U ( n, m ) → U ( n, m ) /U ( m ) , so | b R h ∗ ˆ π ∗ x | = | ˆ π ∗ R h ∗ x | = | R h ∗ x | = | x | = | ˆ π ∗ x | . Consider another bundle U ( n ) → U ( n, m ) /U ( m ) π −→ D n,m and a metric structure on D n,m , by regarding D n,m as the space of orbitsof U ( n, m ) /U ( m ) obtained from the action of a subgroup { b R h | h ∈ U ( n ) } of the isometry group of U ( n, m ) /U ( m ) , such that its projection π is aRiemannian submersion. In fact, for each g ∈ U ( n, m ) ,g (cid:0) U ( n ) × U ( m ) (cid:1) = [ { b R h (cid:0) gU ( m ) (cid:1) (cid:12)(cid:12) h ∈ U ( n ) } , which enables us to identify D n,m with the space of orbits through π ( gU ( m )) = g (cid:0) U ( n ) × U ( m ) (cid:1) , and then we get π ◦ ˆ π = ˜ π. For each g ∈ U ( n, m ) and for each h ∈ U ( n ) , it is obvious that both b L g and b R h preserve the fibers of the bundle π : U ( n, m ) /U ( m ) → D n,m . Infact, π ◦ b R h = π by definition of π and π ◦ b L g = L g ◦ π from( π ◦ b L g ) ◦ ˆ π = ( π ◦ ˆ π ) ◦ L g = ˜ π ◦ L g = L g ◦ ˜ π = ( L g ◦ π ) ◦ ˆ π. Note that we have given two metric structures on D n,m . In other words,we think of two Riemannian manifolds: the first one is the Riemannianmanifold ( D n,m , h· , ·i ) with ˜ π : U ( n, m ) → ( D n,m , h· , ·i ) a Riemanniansubmersion and the other one is the Riemannian manifold ( D n,m , h· , ·i )with π : U ( n, m ) /U ( m ) → ( D n,m , h· , ·i ) a Riemannian submersion. But,˜ π = π ◦ ˆ π and ˆ π ◦ L g − = b L g − ◦ ˆ π, ∀ g ∈ U ( n, m )say that these two metric structures are same, that is, | ˜ π ∗ x | = | ˜ π ∗ x | for any horizontal vector x with respect to the Riemannian submersion˜ π : U ( n, m ) → ( D n,m , h· , ·i ) . To show it, assume x ∈ T g U ( n, m ) , g ∈ U ( n, m ) . The identification of the tangent space of U ( n, m ) at the Iden-tity and u ( n, m ) gives L g − ∗ x ⊥ (cid:0) u ( n ) + u ( m ) (cid:1) and so L g − ∗ x ⊥ u ( m ) , which means that L g − ∗ x is horizontal with respect to ˆ π : U ( n, m ) → U ( n, m ) /U ( m ) . Then, since b L g − : U ( n, m ) /U ( m ) → U ( n, m ) /U ( m ) is an HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m isometry, ˆ π ◦ L g − = b L g − ◦ ˆ π says that x is also horizontal with respect toˆ π : U ( n, m ) → U ( n, m ) /U ( m ) , in other words, | x | = | ˆ π ∗ x | . More precisely, | x | = | L g − ∗ x | = | ˆ π ∗ L g − ∗ x | = | b L g − ∗ ˆ π ∗ x | = | ˆ π ∗ x | . Furthermore, L g − ∗ x ⊥ (cid:0) u ( n ) + u ( m ) (cid:1) also says that L g − ∗ x ⊥ u ( n ) and L g − ∗ x ⊥ u ( m ) at the same time, which means ˆ π ∗ L g − ∗ x = L g − ∗ x + u ( m )is perpendicular to { V + u ( m ) | V ∈ u ( n ) } at the origin of U ( n, m ) /U ( m )from u ( n ) ⊥ u ( m ) and so horizontal with respect to π : U ( n, m ) /U ( m ) → ( D n,m , h· , ·i ) because { V + u ( m ) | V ∈ u ( n ) } is the kernel of π ∗ of thebundle π : U ( n, m ) /U ( m ) → ( D n,m , h· , ·i ) at the origin of U ( n, m ) /U ( m )from [ u ( n ) , u ( m )] = 0 . Then, since b L g : U ( n, m ) /U ( m ) → U ( n, m ) /U ( m )is an isometry preserving the fibers of the bundle π : U ( n, m ) /U ( m ) → D n,m , b L g ∗ ˆ π ∗ L g − ∗ x is also horizontal with repect to π : U ( n, m ) /U ( m ) → ( D n,m , h· , ·i ) . And from b L g ∗ ˆ π ∗ L g − ∗ x = ˆ π ∗ L g ∗ L g − ∗ x = ˆ π ∗ x, we get that ˆ π ∗ x is horizontal with repect to π : U ( n, m ) /U ( m ) → ( D n,m , h· , ·i ) and that | ˆ π ∗ x | = | π ∗ ˆ π ∗ x | , Therefore, ˜ π = π ◦ ˆ π gives | ˜ π ∗ x | = | x | = | ˆ π ∗ x | = | π ∗ ˆ π ∗ x | = | ˜ π ∗ x | . Thus, we will not distinguish one metric on D n,m from the other one. Lemma 2.1.
Given a nontrivial X ∈ M m × n ( C ) , the Lie subalgebra of u ( n, m ) , generated by b X = (cid:18) O n X ∗ X O m (cid:19) , c iX = (cid:18) O n − iX ∗ iX O m (cid:19) , is Span R { e V k , e X k , f iX k (cid:12)(cid:12) k = 1 , , · · · } , where e V k = (cid:18) i ( X ∗ X ) k OO − i ( XX ∗ ) k (cid:19) , e X k = (cid:18) O n ( X ∗ X ) k − X ∗ X ( X ∗ X ) k − O m (cid:19) , and f iX k = (cid:18) O n − i ( X ∗ X ) k − X ∗ iX ( X ∗ X ) k − O m (cid:19) . Furthermore, for q = rk X, Span R { e V k (cid:12)(cid:12) k = 1 , · · · } = Span R { e V k (cid:12)(cid:12) k = 1 , · · · , q } , (2–1) is, at most, a q -dimensional subalgebra of u ( n ) + u ( m ) . Proof.
The first assertion can be given by direct calculations.For the second one, consider a bundle U ( n ) × U ( m ) → U ( n, m ) ˜ π −→ D n,m . For X ∈ M m × n ( C ) , the following Lemma 2.2 make us consider threematrices A ∈ U ( n ) , B ∈ U ( m ) and Σ ∈ M m × n ( C ) such that X = B Σ A ∗ . Then X ∗ X = A Σ ∗ Σ A ∗ , XX ∗ = B ΣΣ ∗ B ∗ and e V k = (cid:18) A OO B (cid:19) (cid:18) i (Σ ∗ Σ) k OO − i (ΣΣ ∗ ) k (cid:19) (cid:18) A ∗ OO B ∗ (cid:19) , where Σ ∗ Σ = diag[ σ , · · · , σ q , , · · · , | {z } ( n − q ) times ] ∈ M n × n and ΣΣ ∗ = diag[ σ , · · · , σ q , , · · · , | {z } ( m − q ) times ] ∈ M m × m for σ ≥ · · · ≥ σ q > , the positive square roots of the decreasingly orderednonzero eigenvalues of W W ∗ , which are the same as the decreasingly orderednonzero eigenvalues of W ∗ W. Thus, the equation (2–1) is trivially obtainedand so it is obvious that its dimension is less that or equal to q. And it is aLie algebra from [ e V k , e V j ] = 0 for k, j = 1 , , · · · . (cid:3) The following Lemma on
Singular value decomposition plays an importantrole in this paper.
Lemma 2.2. [4]
Let W ∈ M m × n ( C ) be given, put a = min { m, n } , andsuppose that rk W = q. (i) There are unitary matrices A ∈ U ( n ) and B ∈ U ( m ) and a squarediagonal matrix Σ a = σ . . . σ a such that σ ≥ σ ≥ · · · ≥ σ q > σ q +1 = · · · = σ a and W = B Σ A ∗ , inwhich Σ = Σ a in case of n = m, (Σ a O ) T ∈ M m × n in case of n < m, (Σ a O ) ∈ M m × n in case of n > m. (ii) The parameters σ , · · · , σ q are the positive square roots of the de-creasingly ordered nonzero eigenvalues of W W ∗ , which are the same as thedecreasingly ordered nonzero eiganvalues of W ∗ W. HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m Recall the following proposition, which gives a sufficient condition to de-termine whether a given complex surface S in D n,m related to ( X, g ) istotally geodesic or not.
Proposition 2.3. [5]
Let ( G, H, σ ) be a symmetric space and g = h + m thecanonical decomposition. Then there is a natural one-to-one correspondencebetween the set of linear subspaces m ′ of m such that [[ m ′ , m ′ ] , m ′ ] ⊂ m ′ andthe set of complete totally geodesic submanifolds M ′ through the origin ofthe affine symmetric space M = G/H, the correspondence being given by m ′ = T ( M ′ ) . Holonomy displacement in the bundle U ( n ) → U ( n, /U ( n ) → U ( n, / ( U ( n ) × U (1))Even though the result in this section can be obtained in view of Corollary1.7, we deal with this section in the way which will be used in Section 5.Given X ∈ M × n ( C ) ∼ = C n , considerSpan R { b X, c iX } = m ′ ⊂ m ⊂ u ( n, m ) . Then, for λ = √ XX ∗ = | b X | and for W = λ X, c W = λ b X = (cid:18) O n W ∗ W (cid:19) , c iW = λ c iX = (cid:18) O n − iW ∗ iW (cid:19) , b V := (cid:18) iW ∗ W OO − i (cid:19) , which generate a 3-dimensional Lie algebra b g with b G its Lie group such that[ c W , c iW ] = 2 b V , [ b V , c W ] = − c iW , [ b V , c iW ] = 2 c W . (3–1)Since
W W ∗ = 1 , Lemma 2.2 says that, for Σ = (1 0 · · · ∈ M × n ( C ) , thereare A ∈ U ( n ) and µ ∈ R such that W = B Σ A ∗ ∈ M × n ( C ) , where B = ( e iµ ) ∈ U (1) , and then forΩ = A ...00 · · · e iµ , we get another expressios for c W , c iW , b V through Ad Ω : u ( n, m ) → u ( n, m ) , c W = Ω O n ...01 0 · · · Ω − (3–2) c iW = Ω − i O n ...0 i · · · Ω − (3–3)and b V = Ω i · · · · · · · · · · · · − i Ω − . (3–4)In fact, W W ∗ = 1 implies that the nonzero eigenvalue of W ∗ W consists ofa simple 1 from Lemma 2.2. More concretely, note that b V = (cid:18) iW ∗ W OO − iW W ∗ (cid:19) = (cid:18) iW ∗ W OO − i (cid:19) and W ∗ W = A Σ ∗ Σ A ∗ = A (cid:18) OO O n − (cid:19) A ∗ . Consider a complex surface S related to ( X, e ) in D n, , which is totallygeodesic from Equation (3–1) and from Proposition 2.3, where e is the iden-tity of U ( n, m ) . For a smooth, simple, closed, orientaion-preserving curve c : [0 , → S , assume that ˆ c : [0 , → U ( n, /U (1) , one of its horizontallifts, is given. Let Area( c ) denote the area of the region on the surface S surrounded by c with respect to the area form ω = ω ( X,e ) related to ( X, e )on S . Put A ( c ) denote the area with respect to the induced metric on S from the metric on D n,m , i.e., the area with respect to the induced area form ω on S from the metric on D n,m . Note that ω = ω from Proposition 1.4. Claim ) There exists an element Ψ ∈ u ( n ) such thatˆ c (1) = ˆ c (0) · e Ψ and Tr(Ψ) = 2 i Area( c ) = 2 iA ( c ) . Consider a curve z ( t ) = r ( t ) e iθ ( t ) in C and another one ˜ γ ( t ) in ˜ S = { e d zW | z ∈ C } ⊂ U ( n,
1) such that ˜ γ ( t ) = e \ z ( t ) W is a lifting of c. Then for ahorizontal lifting of ˆ c, under the identification of u ( n ) and u ( n ) + { } , we HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m can find a curveΨ( t ) = iφ ( t ) W ∗ W = (cid:18) iφ ( t ) W ∗ W OO (cid:19) = Ω iφ ( t ) 0 · · · · · · · · · · · · Ω − ∈ u ( n )such that ˆ c ( t ) = (cid:0) ˜ γ ( t ) U (1) (cid:1) · e Ψ( t ) = (˜ γ ( t ) e Ψ( t ) ) U (1) . Note that from \ z ( t ) W = Ω r ( t ) e − iθ ( t ) O n ...0 r ( t ) e iθ ( t ) · · · Ω − , we get ˜ γ ( t ) = Ω cosh ( r ( t )) 0 · · · e − iθ ( t ) sinh ( r ( t ))0 0... I n − ...0 0 e iθ ( t ) sinh ( r ( t )) 0 · · · r ( t )) Ω − . Then for ¯ c ( t ) = ˜ γ ( t ) e Ψ( t ) , ¯ c ( t ) = Ω e iφ ( t ) cosh ( r ( t )) 0 · · · e − iθ ( t ) sinh ( r ( t ))0 0... I n − ...0 0 e i ( θ ( t )+ φ ( t )) sinh ( r ( t )) 0 · · · r ( t )) Ω − and from L ¯ c ( t ) − ∗ ˙¯ c ( t ) + u (1)= e − Ψ( t ) ˜ γ ( t ) − (cid:0) ˜ γ ′ ( t ) e Ψ( t ) + ˜ γ ( t ) e Ψ( t ) Ψ ′ ( t ) (cid:1) + u (1)= (cid:0) e − Ψ( t ) ˜ γ ( t ) − ˜ γ ′ ( t ) e Ψ( t ) + Ψ ′ ( t ) (cid:1) + u (1) , ˆ c ( t ) = ¯ c ( t ) U (1) is horizontal if and only if the first ( n × n ) − block of e − Ψ( t ) ˜ γ ( t ) − ˜ γ ′ ( t ) e Ψ( t ) + Ψ ′ ( t ) is a zero matrix, in other words, i (cid:0) φ ′ ( t ) − θ ′ ( t ) sinh ( r ( t )) (cid:1) = 0 , i.e., φ ′ ( t ) = θ ′ ( t ) sinh ( r ( t )) . So, for Ψ := Ψ(1) − Ψ(0) ∈ Span R { i ( X ∗ X ) } ∈ u ( n ) and for the region D ( ⊂ S ) enclosed by the given orientation curve c, Proposition 1.4 says thatTr (cid:0) Ψ (cid:1) = i ( φ (1) − φ (0))= i Z θ ′ ( t ) sinh ( r ( t )) dt = 2 i Z [0 ,
1] 12 sinh ( r ( t )) θ ′ ( t ) dt = 2 i Z c sinh r dθ = 2 i Z D d (cid:16) sinh r dθ (cid:17) = 2 i Z D ω = 2 i Area( c )= 2 i A ( c ) . Since ˜ γ (0) = ˜ γ (1) and [Ψ(0) , Ψ(1)] = O n in u ( n ) , we get¯ c (0) − ¯ c (1) = (cid:0) ˜ γ (0) e Ψ(0) (cid:1) − (cid:0) ˜ γ (1) e Ψ(1) (cid:1) = e − Ψ(0) e Ψ(1) = e Ψ and so ˆ c (1) = ¯ c (1) K = (¯ c (0) e Ψ ) K = ¯ c (0) K · e Ψ = ˆ c (0) · e Ψ , i.e. , the holonomy displacement is given by the right action of e Ψ ∈ U ( n ) . Proof of Proposition 1.4
Proof.
For the part (i), let W = (cid:12)(cid:12) b X (cid:12)(cid:12) X, c W = \ (cid:12)(cid:12) b X (cid:12)(cid:12) X = (cid:12)(cid:12) b X (cid:12)(cid:12) b X, and c iW = \ (cid:12)(cid:12) b X (cid:12)(cid:12) iX = (cid:12)(cid:12) b X (cid:12)(cid:12) c iX and consider an ordered pair (cid:0)c W , c iW (cid:1) , which induces an oriented orthonor-mal basis of each tangent space of S and an area form ω = ω ( X,e ) relatedto ( X, e ) on S , where e is the identity of U ( n, m ) . Assume rk W = q, min { n, m } = a and Max { n, m } = b. Then, Lemma 2.2says that there are nonnegative real numbers σ ≥ · · · ≥ σ q > σ q +1 = · · · = σ a = · · · = σ b = · · · = σ n + m such that σ , · · · , σ q are positive square roots of nonzero eigenvalues of W W ∗ and W ∗ W and that Σ a = diag[ σ , · · · , σ a ] and W = B Σ A ∗ , A ∈ U( n ) , B ∈ U( m ) , HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m where Σ = Σ a in case of n = m, (Σ a O ) T ∈ M m × n in case of n < m, (Σ a O ) ∈ M m × n in case of n > m. For r ≥ θ ∈ R , let z = re iθ and consider ˜ S = { e d zW | z ∈ C } . LetΩ = (cid:18)
A OO B (cid:19) . Then, e d zW = P ∞ k =0 1(2 k )! r k ( W ∗ W ) k e − iθ P ∞ k =0 1(2 k +1)! r k +1 ( W ∗ W ) k W ∗ e iθ P ∞ k =0 1(2 k +1)! r k +1 W ( W ∗ W ) k P ∞ k =0 1(2 k )! r k ( W W ∗ ) k ! = (cid:18) A OO B (cid:19) (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) (cid:18) A ∗ OO B ∗ (cid:19) = Ω (cid:18) Γ n ( z ) Λ( z ) ∗ Λ( z ) Γ m ( z ) (cid:19) Ω ∗ from W ( W ∗ W ) k = ( W W ∗ ) k W, W ∗ W = A Σ ∗ Σ A ∗ and W W ∗ = B ΣΣ ∗ B ∗ , where Γ n ( z ) = diag[cosh ( σ r ) , · · · , cosh ( σ n r )] , Γ m ( z ) = diag[cosh ( σ r ) , · · · , cosh ( σ m r )] , and for Λ a ( z ) = diag[ e iθ sinh ( σ r ) , · · · , e iθ sinh ( σ a r )] , Λ( z ) = Λ a ( z ) in case of a = n = m, (Λ a ( z ) O ) T ∈ M m × n in case of a = n < m, (Λ a ( z ) O ) ∈ M m × n in case of a = m < n. Note z e d zW : C → ˜ S is a bijection.By abusing notations, we use the same letter ( r, θ ) on S for the inducedone from the coordinate chart ( r, θ ) on ˜ S. Then, the relation˜ π ∗ ∂∂r = ∂∂r ◦ ˜ π and ˜ π ∗ ∂∂θ = ∂∂θ ◦ ˜ π (4–1)holds. Note that, under the identification of the Lie algebra u ( n, m ) and T e G, where G = U ( n, m ) , the direct calculation shows that L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW = (cid:18) O e − iθ W ∗ e iθ W O (cid:19) , (4–2)and L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (4–3)= − i P ∞ k =0 1(2 k )! (2 r ) k ( W ∗ W ) k − i e − iθ P ∞ k =0 1(2 k +1)! (2 r ) k +1 ( W ∗ W ) k W ∗ i e iθ P ∞ k =0 1(2 k +1)! (2 r ) k +1 W ( W ∗ W ) k i P ∞ k =0 1(2 k )! (2 r ) k ( W W ∗ ) k ! , so for W ∗ W = A Σ ∗ Σ A ∗ = A Σ ∗ n Σ n A ∗ , where Σ n = diag[ σ , · · · , σ n ] , we get ω (cid:16) ∂∂r | ˜ π ( e d zW ) , ∂∂θ | ˜ π ( e d zW ) (cid:17) = det h L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW , c W i h L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW , c W ih L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW , c iW i h L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW , c iW i ! = det cos θ − sin θ P ∞ k =0 1(2 k +1)! (2 r ) k +1 Tr( W ∗ W ) k +1 sin θ cos θ P ∞ k =0 1(2 k +1)! (2 r ) k +1 Tr( W ∗ W ) k +1 ! = 12 ∞ X k =0 k + 1)! (2 r ) k +1 ( σ k +1)1 + · · · + σ k +1) n )= 12 n X j =1 σ j sinh (2 σ j r )= n X j =1 σ j sinh ( σ j r ) cosh ( σ j r ) . Therefore, ω = n X j =1 σ j cosh ( σ j r ) sinh ( σ j r ) dr ∧ dθ = d (cid:16) n X j =1 12 sinh ( σ j r ) dθ (cid:17) . Furthermore, c W ∗ c W = (cid:18) W ∗ W OO W W ∗ (cid:19) shows that 1 = | c W | = Tr( W ∗ W ) = Tr(Σ ∗ n Σ n ) = σ + · · · + σ n . Before proving the part (ii), note that the restriction of ˜ π on ˜ S is abijection to S from Remark 1.11, which induces that the restriction of L g on S is a bijection to S from ˜ π ◦ L g = L g ◦ ˜ π. So, for L g − , its restriction L g − : S → S is also a bijection.To prove the part (ii), let ( r , θ ) be a coordinate chart on S = ˜ π ( g ˜ S )with a complex structure induced from ˜ S, i.e., r (cid:0) ˜ π ( ge d zW ) (cid:1) = r (cid:0) e d zW (cid:1) and θ (cid:0) ˜ π ( ge d zW ) (cid:1) = θ (cid:0) e d zW (cid:1) , which says r ◦ ˜ π ◦ L g = r and θ ◦ ˜ π ◦ L g = θ. Then, from the equations (4–1), ∂∂r (cid:12)(cid:12) ˜ π ◦ L g ( e d zW ) = (cid:0) ˜ π ◦ L g (cid:1) ∗ ∂∂r (cid:12)(cid:12) e d zW = (cid:0) L g ◦ ˜ π (cid:1) ∗ ∂∂r (cid:12)(cid:12) e d zW = L g ∗ ∂∂r (cid:12)(cid:12) ˜ π ( e d zW ) , HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m so we get ∂∂r (cid:12)(cid:12) ˜ π ( e d zW ) = L g − ∗ ∂∂r (cid:12)(cid:12) ˜ π ( ge d zW ) . Similarly, we also get ∂∂θ (cid:12)(cid:12) ˜ π ( e d zW ) = L g − ∗ ∂∂θ (cid:12)(cid:12) ˜ π ( ge d zW ) . Then, ω ( X,g ) (cid:16) ∂∂r (cid:12)(cid:12) ˜ π ( ge d zW ) , ∂∂θ (cid:12)(cid:12) ˜ π ( ge d zW ) (cid:17) = ω ( X,g ) (cid:16) ˜ π ∗ L g ∗ ∂∂r (cid:12)(cid:12) e d zW , ˜ π ∗ L g ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:17) = det h L ( ge d zW ) − ∗ L g ∗ ∂∂r (cid:12)(cid:12) e d zW , c W i h L ( ge d zW ) − ∗ L g ∗ ∂∂θ (cid:12)(cid:12) e d zW , c W ih L ( ge d zW ) − ∗ L g ∗ ∂∂r (cid:12)(cid:12) e d zW , c iW i h L ( ge d zW ) − ∗ L g ∗ ∂∂θ (cid:12)(cid:12) e d zW , c iW i ! = det h L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW , c W i h L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW , c W ih L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW , c iW i h L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW , c iW i ! = ω (cid:16) ∂∂r (cid:12)(cid:12) ˜ π ( e d zW ) , ∂∂θ (cid:12)(cid:12) ˜ π ( e d zW ) (cid:17) = ω (cid:16) L g − ∗ ∂∂r (cid:12)(cid:12) ˜ π ( ge d zW ) , L g − ∗ ∂∂θ (cid:12)(cid:12) ˜ π ( ge d zW ) (cid:17) = L g − ∗ ω (cid:16) ∂∂r (cid:12)(cid:12) ˜ π ( ge d zW ) , ∂∂θ (cid:12)(cid:12) ˜ π ( ge d zW ) (cid:17) . To prove part (iii), consider the induced area form ω on S from the metricon D n,m . Since the restriction L g − : S → S of L g − : D n,m → D n,m is anisometry, we get ω = L ∗ g − ω for the induced area form ω on S from the metric on D n,m . And, underthe notation x h of the horizontal part of a tangent vector x ∈ T (cid:0) U ( n, m ) (cid:1) with respect to the Riemannian submersion ˜ π : U ( n, m ) → D n,m , Equations(4–2) and (4–3) show that L e − d zW ∗ (cid:16) ∂∂r (cid:12)(cid:12) e d zW (cid:17) h = (cid:16) L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW (cid:17) h = L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW and that L e − d zW ∗ (cid:16) ∂∂θ (cid:12)(cid:12) e d zW (cid:17) h = (cid:16) L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:17) h = O n − i e − iθ P ∞ k =0 (2 r ) k +1 (2 k +1)! ( W ∗ W ) k W ∗ i e iθ P ∞ k =0 (2 r ) k +1 (2 k +1)! W ( W ∗ W ) k O m = O n − i e − iθ P ∞ k =0 (2 r ) k +1 (2 k +1)! A (Σ ∗ Σ) k Σ ∗ B ∗ i e iθ P ∞ k =0 (2 r ) k +1 (2 k +1)! B Σ(Σ ∗ Σ) k A ∗ O m = Ω (cid:18) O n − i e − iθ Υ( z ) ∗ i e iθ Υ( z ) O m (cid:19) Ω ∗ , where Υ a ( z ) = diag[sinh (2 σ r ) · · · , sinh (2 σ a r )] , and Υ( z ) = Υ a in case of n = m, (Υ a O ) T ∈ M m × n in case of n < m, (Υ a O ) ∈ M m × n in case of n > m. Then, (cid:12)(cid:12)(cid:12) L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW (cid:12)(cid:12)(cid:12) = r
12 Re (cid:16) Tr (cid:16)(cid:0) L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW (cid:1) ∗ L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW (cid:17)(cid:17) = q σ + · · · + σ n = 1 , h L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW , L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW i = Re (cid:16) Tr (cid:16)(cid:0) L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW (cid:1) ∗ L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:17)(cid:17) = Re (cid:16) Tr (cid:16) Ω (cid:18) O n e − iθ Σ ∗ e iθ Σ O m (cid:19) (cid:18) O n − i e − iθ Υ( z ) ∗ i e iθ Υ( z ) O m (cid:19) Ω ∗ (cid:17)(cid:17) = Re (cid:16) Tr (cid:16) (cid:18) i Σ ∗ Υ( z ) OO − i ΣΥ( z ) ∗ (cid:19) (cid:17)(cid:17) = 0 HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m and (cid:12)(cid:12)(cid:12)(cid:16) L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:17) h (cid:12)(cid:12)(cid:12) = r
12 Re (cid:16) Tr (cid:16)(cid:16)(cid:0) L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:1) h (cid:17) ∗ (cid:0) L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW (cid:1) h (cid:17)(cid:17) = s
18 Re (cid:16) Tr (cid:16) (cid:18) Υ( z ) ∗ Υ( z ) OO Υ( z )Υ( z ) ∗ (cid:19) (cid:17)(cid:17) = 12 vuut n X j =1 sinh (2 σ j r ) , which show the former one of the part (iii).To show the latter one, assume σ = · · · = σ q > σ q +1 = · · · σ n . Then, σ = √ q from σ + · · · + σ n = 1 and ω = √ q sinh (cid:0) √ q r (cid:1) dr ∧ dθ = √ q sinh (cid:0) √ q r (cid:1) cosh (cid:0) √ q r (cid:1) dr ∧ dθ = d (cid:16) q sinh (cid:0) √ q r (cid:1)(cid:17) dθ = d (cid:16) n X j =1 12 sinh ( σ j r ) dθ (cid:17) = ω . And W ∗ W = A Σ ∗ Σ A ∗ = A diag[ σ , · · · σ , | {z } q times , · · · | {z } ( n − q ) times ] A ∗ shows thatdet( xI n − W ∗ W ) = det (cid:0) A ( xI n − diag[ σ , · · · σ , | {z } q times , · · · | {z } ( n − q ) times ] ) A ∗ (cid:1) = ( x − σ ) q x ( n − q ) . Proof of Theorem 1.5
We follow the notation in the proof of Proposition 1.4.Let K = U ( m ) . Note that the equations˜ π = π ◦ ˆ π, ˆ π ◦ L g = b L g ◦ ˆ π and ˜ π ◦ L g = L g ◦ ˜ π give π (cid:0) b L g (ˆ π ( ˜ S )) (cid:1) = S = L g ( S ) . And for any g ∈ U ( n, m ) and for any h ∈ U ( n ) , both b L g , b R h : U ( n, m ) /U ( m ) → U ( n, m ) /U ( m ) are isometries preserving the fibers of thebundle π : U ( n, m ) /U ( m ) → D n,m such that π ◦ b L g = L g ◦ π and π ◦ b R h = π. Thus the curve c on S can be written by L g ◦ c for some curve c on S , and then for a horizontal lift ˆ c of c on U ( n, m ) /U ( m ) , the horizontal lift ˆ c of c = L g ◦ c will be given by b R h ◦ b L g ◦ ˆ c for some h ∈ U ( n ) . Note that ifˆ c (1) = ˆ c (0) · h , h ∈ U ( n )then ( b R h ◦ b L g ◦ ˆ c )(1) = ( b R h ◦ b L g ◦ ˆ c )(0) · (cid:0) h − h h (cid:1) . And, if h = e Φ for some Φ ∈ u ( n ) , then h − h h = e Ad h − Φ and Tr(Ad h − Φ) = Tr(Φ) . So, Proposition 1.4(ii) enables us to suppose that g is the identity of U ( n, m )without loss of generality.Before constructing the horizontal lift ˆ c ( t ) of c ( t ) ∈ S , t ∈ [0 , , in thebundle U ( n, m ) /U ( m ) π → D n,m , let z ( t ) = r ( t ) e iθ ( t ) and ˜ γ ( t ) = e \ z ( t ) W , the lift of c in the bundle U ( n, m ) → D n,m , lying in ˜ S ⊂ U ( n, m ) . FromLemma 2.1, consider the Lie subgroup b G (cid:0) ⊂ U ( n, m ) (cid:1) of the Lie algebra ˆ g generated by { c W , c iW } , which is the same as the one by { b X, c iX } . Followthe notation of Lemma 2.1 and let b H be the Lie group of the Lie algebra ˆ h generated by { e V k | k = 1 , , · · · } . Then b H is a subset of the intersection of b G and U ( n ) × U ( m ) , which means that for each element of b H, the map, givenby its right multiplication in b G with the left invariant metric induced fromthe metric of U ( n, m ) determined by (1–1), is an isometry of b G. Considerthe inclusion map ι : b G ֒ → U ( n, m ) , its Lie algebra homomorphism dι : ˆ g → u ( n, m ) and the Riemannian submersion b pr : b G → b G/ b H .
Since dι ( e V k ) = e V k , dι ( e X k ) = e X k and dι ( f iX k ) = f iX k (5–1)for k = 1 , , · · · , Lemma 2.1, the equation (4–2) on L e − d zW ∗ ∂∂r (cid:12)(cid:12) e d zW and theequation (4–3) on L e − d zW ∗ ∂∂θ (cid:12)(cid:12) e d zW in the proof of Proposition 1.4 show that ˜ S is a subset of b G. The tangent vector of any horizontal curve ˜ c in b G → b G/ b H is spanned by e X k and by f iX k , k = 1 , , · · · . The tangent vector of itsinduced curve ι ◦ ˜ c in U ( n, m ) is too, from the equation (5–1), which meansthat, by abusing of notation, ι ◦ ˜ c = ˜ c is a horizontal curve in the bundle U ( n, m ) → D n,m since both e X k and f iX k are still horizontal in this bundle. HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m In fact, e V k , e X k and f iX k in g are different from e V k , e X k and f iX k in ˆ g as tangentvector fields, but they are ι -related, i.e., ι ∗ e V k = e V k ◦ ι, ι ∗ e X k = e X k ◦ ι and ι ∗ f iX k = f iX k ◦ ι, and so ˙ z}|{ ι ◦ ˜ c = ι ∗ ˙˜ c is generated by e X k , f iX k in g . And the horizontal curve ˜ c in b G with b pr ◦ ˜ c = b pr ◦ ˜ γ can be given by ˜ c ( t ) = ˜ γ ( t ) e b Ψ( t ) for some curve b Ψ : [0 , → ˆ h and will be regarded as a horizontal lift of c in the bundle U ( n, m ) → D n,m . For b Ψ( t ) = (cid:18) η ( t ) OO η ( t ) (cid:19) ∈ ˆ h ⊂ u ( n ) + u ( m ) , let Ψ( t ) = (cid:18) η ( t ) OO O m (cid:19) ∈ u ( n ) + { O m } . Then, for the curve ¯ c ( t ) = ˜ γ ( t ) e Ψ( t ) in U ( n, m ) , the equation˜ c ( t ) K = ˜ γ ( t ) e b Ψ( t ) K = ˜ γ ( t ) e Ψ( t ) K = ¯ c ( t ) K holds. Therefore, to construct the horizontal lift ˆ c ( t ) = ˜ c ( t ) K = ¯ c ( t ) K of c ( t ) in the bundle U ( n, m ) /U ( m ) π → D n,m , it suffices to find such a curveΨ( t ) ∈ u ( n ) . Note that ˜ γ ( t ) = (cid:18) A OO B (cid:19) (cid:18) Γ n ( t ) Λ( t ) ∗ Λ( t ) Γ m ( t ) (cid:19) (cid:18) A ∗ OO B ∗ (cid:19) (5–2) = Ω (cid:18) Γ n ( t ) Λ( t ) ∗ Λ( t ) Γ m ( t ) (cid:19) Ω ∗ , (5–3)where Γ j ( t ) = diag[cosh ( σ r ( t )) , · · · , cosh ( σ j r ( t ))] , j = n, m and for Λ a ( t ) = diag[ e iθ ( t ) sinh ( σ r ( t )) , · · · , e iθ ( t ) sinh ( σ a r ( t ))] , Λ( t ) = Λ a ( t ) in case of a = n = m, (Λ a ( t ) O ) T ∈ M m × n in case of a = n < m, (Λ a ( t ) O ) ∈ M m × n in case of a = m < n. Lemma 2.1, W ∗ W = (cid:12)(cid:12) b X (cid:12)(cid:12) X ∗ X and W W ∗ = (cid:12)(cid:12) b X (cid:12)(cid:12) XX ∗ enable us to considera curve b Ψ( t ) = (cid:18) i P qk =1 φ k ( t )( W ∗ W ) k OO − i P qk =1 φ k ( t )( W W ∗ ) k (cid:19) ∈ ˆ h such that q = rk W and that ˜ c ( t ) = ˜ γ ( t ) e b Ψ( t ) . Let Ψ( t ) = (cid:18) i P qk =1 φ k ( t )( W ∗ W ) k OO O m (cid:19) ∈ u ( n ) . Then, by abusing of notations,we can say that Ψ( t ) ∈ Span R { i ( X ∗ X ) k } qk =1 . And Ψ( t ) = Ω diag h i q X k =1 σ k φ k ( t ) , · · · , i q X k =1 σ kn φ k ( t ) , , · · · , | {z } m -times i Ω ∗ . So, we get e Ψ( t ) = Ω diag h e i P qk =1 σ k φ k ( t ) , · · · , e i P qk =1 σ kn φ k ( t ) , , · · · , | {z } m -times i Ω ∗ (5–4)and ˆ c ( t ) = ˜ γ ( t ) e b Ψ( t ) K = ˜ γ ( t ) e Ψ( t ) K = ¯ c ( t ) K for ¯ c ( t ) = ˜ γ ( t ) e Ψ( t ) . Note L ¯ c ( t ) − ∗ ˙¯ c ( t ) + u ( m )= e − Ψ( t ) ˜ γ ( t ) − (cid:0) ˜ γ ′ ( t ) e Ψ( t ) + ˜ γ ( t ) e Ψ( t ) Ψ ′ ( t ) (cid:1) + u ( m )= (cid:0) e − Ψ( t ) ˜ γ ( t ) − ˜ γ ′ ( t ) e Ψ( t ) + Ψ ′ ( t ) (cid:1) + u ( m ) , and the direct calculation e − Ψ( t ) ˜ γ ( t ) − ˜ γ ′ ( t ) e Ψ( t ) + Ψ ′ ( t ) through the singularvalue decompositions (5–2) and (5–4) of ˜ γ and e Ψ( t ) says that its first ( n × n )-block matrix is A diag h i (cid:16) q X k =1 σ kj φ ′ k ( t ) − θ ′ ( t ) sinh ( σ j r ( t )) (cid:17)i nj =1 A ∗ . Note that σ q +1 = · · · = σ n = 0 if n > q = rk W. Then ˆ c ( t ) = ¯ c ( t ) K is a horizontal curve in G/K if and only if q X k =1 σ kj φ ′ k ( t ) = θ ′ ( t ) sinh ( σ j r ( t )) , j = 1 , · · · , q. Therefore, Proposition 1.4 says that, for Ψ := Ψ(1) − Ψ(0) ∈ Span R { i ( X ∗ X ) k } qk =1 ⊂ u ( n ) and for the region D ( ⊂ S ) enclosed by the HE HOLONOMY ON THE PRINCIPAL U ( n ) BUNDLE OVER D n,m given orientation-preserving curve c : [0 , → S , Tr (cid:0) Ψ (cid:1) = i n X j =1 q X k =1 σ kj ( φ k (1) − φ k (0))= i Z n X j =1 θ ′ ( t ) sinh ( σ j r ( t )) dt = 2 i Z [0 , n X j =1 12 sinh ( σ j r ( t )) θ ′ ( t ) dt = 2 i Z c n X j =1 12 sinh ( σ j r ) dθ = 2 i Z D d (cid:16) n X j =1 12 sinh ( σ j r ) dθ (cid:17) = 2 i Z D ω = 2 i Area( c ) . And, ˜ γ (0) = ˜ γ (1) and the Lie bracket [ i ( X ∗ X ) k , i ( X ∗ X ) j ] = O n in u ( n ) saythat ¯ c (0) − ¯ c (1) = (cid:0) ˜ γ (0) e Ψ(0) (cid:1) − (cid:0) ˜ γ (1) e Ψ(1) (cid:1) = e − Ψ(0) e Ψ(1) = e Ψ . Since the holonomy displacement is given by the right action of e Ψ ∈ U ( n ) , i.e. , ˆ c (1) = ¯ c (1) K = (¯ c (0) e Ψ ) K = ¯ c (0) K · e Ψ = ˆ c (0) · e Ψ , the theorem is proved. (cid:3) References
1. T. Byun, Y. Choi, The topological aspect of the holonomy displacement on the pricipal U ( n ) bundles over Grassmannian manifiolds, Topology Appl. 196 (2015), 8-21.2. Y. Choi, K.B. Lee, Holonomy displacements in the Hopf bundles over C H n and thecomplex Heisenberg groups, J. Korean Math. Soc. 49, 733–743 (2012)3. D.Gromoll, G.Walschap, Metric foliations and curvature, Progress in mathematics,vol.268, Birkhauser4. R.A.Horn, C.R.Johnson, Matrix Analysis, second edition, Cambridge UniversityPress, 20135. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, Reprintof the 1969 original. Wiley Classics Library, A Wiley-Interscience Publication, JohnWiley and Sons, Inc., New York, 1996.6. U. Pinkall, Hopf tori in S , Invent. Math. 81, 379–386 (1985) Department of Mathematics and Statistics, Sejong University, Seoul 143-747, Korea
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