J. Pseudo-Differ. Oper. Appl. (2015) 6:265–277 DOI 10.1007/s11868-015-0114-z

On the Kähler form of complex L p space and its Lagrangian subspaces Yang Liu1

Received: 14 February 2015 / Revised: 16 April 2015 / Accepted: 18 April 2015 / Published online: 24 May 2015 © Springer Basel 2015

Abstract In this article, we obtain the Kähler forms for complex L p spaces, 1 ≤ p < ∞, and we find and describe explicitly the set of all Lagrangian subspaces of the complex L p space. The results in this article show that the Lagrangians of complex L 2 space are distinct from those of complex L p spaces for 1 ≤ p < ∞, p = 2. As an application, the symplectic structure determined by the Kähler form can be used to determine the symplectic form of the complex Holmes–Thompson volumes restricted on complex lines in integral geometry of complex L p space. Keywords Complex space · Kähler form · Lagrangian subspace · Differential geometry · Complex structure Mathematics Subject Classification

70S05 · 32Q15 · 53C56 · 47A15 · 53C65

1 Introduction As a generalization of Riemannian manifolds, real Finsler spaces have been studied (see for instance [5], and [6,19]). But complex Finsler spaces (see for instance, [1] and [17]) have become an interest of research for the studies of geometry, including differential geometry and integral geometry, in recent decades. As a typical complex Finsler space, complex L p space has the main features of a complex Finsler space. As such, we focus on complex L p space in this paper, but some results can be generalized to general complex Finsler spaces, on which one can refer to [14]. The L p space, as a generalization of Euclidean space, has a rich structure in functional analysis

B 1

Yang Liu [email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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(see for instance, [11] and [25]), and particularly Banach space. Furthermore, it has broad applications in statistics (see for instance [23]), engineering (see for instance [7]), computational science (see for instance [24]), and other areas. Along this way, L p , 0 < p ≤ 1 , in the sense of conjugacy to the scenario of this paper, also has broad applications, in particular, signal processing in engineering, on which one can see [8,12], and [13]. The main contribution of this paper is to give explicit canonical Kähler form for complex L p space and to answer questions on the existence of Lagrangian subspace. Furthermore, we find and describe explicitly the set of Lagrangian subspaces of C2 with complex L p norm, 1 ≤ p < ∞. As an application in integral geometry, the symplectic structure determined by the Kähler form can be used to determine the symplectic form of complex Holmes–Thompson volumes restricted on complex lines in integral geometry of complex L p spaces, since Holmes–Thompson volumes, as measures, depend on the differential structures of the spaces. This paper is structured as follows: in Sect. 2, we give explicitly the Hermitian metric for complex L p plane induced from its norm and the canonical Kähler form κ p on the complex L p plane. Moreover, we describe the set of Lagrangian subspaces of C2 with complex L 2 norm and L 1 norm, 1 ≤ p < ∞; and in Sect. 3, we give the explicit description on the set of Lagrangian subspaces of C2 with complex L p norm, 1 ≤ p < ∞ and p = 2.

2 Kähler forms on complex L p space In this section, we five the explicit Kähler forms for complex L p space, 1 ≤ p < ∞, and characterize the Lagrangian subspaces for complex L 2 space. To obtain the canonical Kähler form for complex L p space, which is equipped with the complex L p norm 1/ p  F p (z, w) = ||(z, w)|| p := |z| p + |w| p

(2.1)

for 1 ≤ p < ∞, we can use the generalized Kähler potential G p := F p2 , and then the Kähler form κ p will be the negative imaginary part of 21 ∂∂(G p ). First, we have ∂(F p2 ) = =

2 p

2

(|z| p + |w| p ) p

2 p p (|z|

2

+ |w| p ) p 2

= (|z| p + |w| p ) p 2

= (1 + | wz | p ) p in other word,

∂ 2 ∂z F p

2

= (1 + | wz | p ) p

−1

−1

−1

−1

−1

∂ (|z| p + |w| p )

∂((zz) p/2 + (ww) p/2 )

(2.2)

(|z| p−2 zdz + |w| p−2 wdw) 2

zdz + (1 + | wz | p ) p

z and

∂ 2 ∂w F p

−1

wdw, 2

= (1 + | wz | p ) p

−1

w.

On the Kähler form of complex L p space and its...

Using ∂ ∂z

∂ 1 p ∂z (| z | )

∂ − 2p ) ∂z ((zz)

=

267 p

1 = (− 2p )(zz)− 2 −1 z = (− 2p ) z|z| p , we get

     p  2 −1 2  2 −1  p w p w p p −1 w ∂ 1+| | 1 +   z = 1 + | |p +z z z ∂z z 2   p  −1  w p = 1 +   z   p  2 −2      w p 1 ∂ 2 − 1 z 1 +   | |p + |w| p p z ∂z z   p  2 −1  2   

w p  w  p p −2 w p p = 1 +   − 1 1 +   + | | z 2 z z  p   p  2 −2   w p p w 1 +   , (2.3) = 1 +   z 2 z

thus

 2   ∂2 w p p −2 p w p = 1+| | 1+ | | . ∂z∂z z 2 z

(2.4)

and by symmetry of w and z in (2.4) we know z 2p −2 p z

∂2 = 1 + | |p 1 + | |p . ∂w∂w w 2 w By

∂ p ∂w (|w| )

∂ ∂w

thus

=

p ∂ 2 ∂w (ww)

=

p p 2 −1 w 2 (ww)

=

(2.5)

p |w| p 2 w ,

   2  2 −2   p w w p p −1 1 p |w| p 2 − 1 z 1 + | |p 1+| | z = z p z |z| p 2 w  2 w p p −2 p z w p | | 1+| | , (2.6) = 1− 2 w z z   2 −2 p ∂2 w p z w p F p2 = 1 − | | 1 + | |p . ∂w∂z 2 w z z

(2.7)

By symmetry of z and w in (2.7), we also have z 2p −2 ∂2 p w z p F p2 = 1 − | | 1 + | |p . ∂z∂w 2 z w w

(2.8)

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So we have the following Theorem 2.1 The Hermitian metric for complex L p plane induced from its norm is hp =



2 −2



  p 1 + | wz | p 1 + 2p | wz | p dz ⊗ dz + 1 − 2p wz | wz | p dw ⊗ dz

  2 −2     1 − 2p wz | wz | p dz ⊗ dw + 1 + 2p | wz | p dw ⊗ dw . + 21 1 + | wz | p p (2.9)

1 2

Let z = x + yi and w = u + vi, then we can express the Kähler form by taking the negative imaginary part of (2.9), which is the following Corollary 2.2 The canonical Kähler form κ p for complex L p plane induced from its norm is

2 −2

p 1 + 2p | wz | p d x ∧ dy κ p = 1 + | wz | p

2 −2     p + 1 − 2p 1 + | wz | p | wz | p Re wz (d x ∧ dv − dy ∧ du)

2 −2     p + 1 − 2p 1 + | wz | p | wz | p I m wz (d x ∧ du + dy ∧ dv)   2 −2   + 1 + | wz | p p 1 + 2p | wz | p du ∧ dv.

(2.10)

One can also take the real part of (2.9), which is a Riemannian metric denoted by g p . The Kähler form κ p combines the Riemannian metric g p and the complex structure J as follows Corollary 2.3 Given any (z 1 , w1 ), (z 2 , w2 ) in complex L p plane, then g p ((z 1 , w1 ), (z 2 , w2 )) = κ p ((z 1 , w1 ), J (z 2 , w2 )).

(2.11)

Proof First, the real part of (2.9) yields

2 −2

p g p = 1 + | wz | p 1 + 2p | wz | p (d x ⊗ d x + dy ⊗ dy)

2 −2     p + 1 − 2p 1 + | wz | p | wz | p Re wz ((du ⊗ d x + dv ⊗ dy) + (d x ⊗ du + dy ⊗ dv))

2 −2     p + 1 − 2p 1 + | wz | p | wz | p I m wz ((du ⊗ dy − dv ⊗ d x)

(2.12)

+ (dy ⊗ du − d x ⊗ dv))  2 −2    1 + 2p | wz | p (du ⊗ du + dv ⊗ dv) . + 1 + | wz | p p We have the following equalities, (d x ⊗d x +dy ⊗dy)((z 1 , w1 ), (z 2 , w2 )) = Re(z 1 z 2 ) = d x ∧dy((z 1 , w1 ), J (z 2 , w2 )) (2.13)

On the Kähler form of complex L p space and its...

269

and ((du ⊗ d x + dv ⊗ dy) + (d x ⊗ du + dy ⊗ dv))((z 1 , w1 ), (z 2 , w2 )) = Re(z 1 w2 + z 2 w1 ) = (d x ∧ dv − dy ∧ du)((z 1 , w1 ), J (z 2 , w2 )), (2.14) similarly, (du⊗du+dv⊗dv)((z 1 , w1 ), (z 2 , w2 )) = Re(w1 w2 ) = du∧dv((z 1 , w1 ), J (z 2 , w2 )) (2.15) and ((du ⊗ dy − dv ⊗ d x) + (dy ⊗ du − d x ⊗ dv))((z 1 , w1 ), (z 2 , w2 )) = I m(z 1 w2 + z 2 w1 ) = (d x ∧ du + dy ∧ dv)((z 1 , w1 ), J (z 2 , w2 )).

(2.16)  

Then comparing (2.10) and (2.12), (2.11) in the claim follows.

Our goal is to get all Lagrangian subspaces of the Kähler form κ p for every 1 < p < ∞. For clearness, we consider the case of complex L 2 and L 1 spaces first, and then generalize them. Since (2.17) κ2 = d x ∧ dy + du ∧ dv from (2.12), we have Theorem 2.4 The set of Lagrangian subspaces of C2 with complex L 2 norm is  T := {span((z 1 , w1 ), (z 2 , w2 )) : z 1 , w1 , z 2 , w2 ∈ C, I m(z 2 z 1 ) = I m(w1 w2 )} . (2.18) Proof Suppose κ2 vanishes on a plane P spanned by (z 1 , w1 ) and (z 2 , w2 ). Since d x ∧ dy((z 1 , w1 ), (z 2 , w2 )) = I m(z 2 z 1 )

(2.19)

du ∧ dv((z 1 , w1 ), (z 2 , w2 )) = I m(w2 w1 )

(2.20)

I m(z 2 z 1 ) + I m(w2 w1 ) = 0,

(2.21)

and we then have i.e. I m(z 2 z 1 ) = I m(w1 w2 ).

 

T for any complex line L in C2 . Moreover, one can Remark 2.5 Obviously L ∈ / T0 ∪   {ϕ(span((1, 0), (0, 1))) : ϕ ∈ U (2)} by normalizing the basis show that T0 ∪ T = {(z 1 , w1 ), (z 2 , w2 )}, on which one can refer to to, for instance, [9,20], and [10].

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However, we have

  κ1 = 1 + 21 | wz | d x ∧ dy + 1 + 21 | wz | du ∧ dv



+ 21 Re wz | wz | (d x ∧ dv − dy ∧ du) + 21 I m wz | wz | (d x ∧ du + dy ∧ dv) . (2.22) Using the equalities (2.19), (2.20), (d x ∧ dv − dy ∧ du)((z 1 , w1 ), (z 2 , w2 )) = I m(w2 z 1 − w1 z 2 )

(2.23)

(d x ∧ du + dy ∧ dv)((z 1 , w1 ), (z 2 , w2 )) = Re(w2 z 1 − w1 z 2 ),

(2.24)

and

we get

    1 w 1 z κ1 ((z 1 , w1 ) , (z 2 , w2 )) = 1 + | | I m (z 2 z 1 ) + 1 + | | I m (w2 w1 ) 2 z 2 w   1 z w (2.25) + Im | | (w2 z 1 − w1 z 2 ) . 2 w z

Now let’s state the result of Lagrangians for L 1 and prove it by analyzing several cases. Theorem 2.6 The set of Lagrangian subspaces of C2 with complex L 1 norm is T2 ∪T1 , where T2 := {span((z, 0), (0, w)) : z, w ∈ U (1)} ∼ (2.26) = U (1) × U (1) and T1 := {P : P = {λ(z, w) : λ ∈ R, z, w ∈ U (1), zw is a constant in U (1)}} ∼ = U (1). (2.27) Proof Firstly we can show that P = {λ(z, w) : λ ∈ R, z, w ∈ U (1), zw is a constant in U (1)}

(2.28)

is identical to some 

P := span



z1, z1e

iθ





z2 z2 , z 2 , 1 2 eiθ |z 1 |

 (2.29)

where z 1 , z 2 ∈ C\{0} . For any λ(eiϕ , eiψ ) ∈ P, let z 1 = λeiϕ , θ = ψ − ϕ, we have z2 z

P = span((z 1 , z 1 eiθ ), (z 2 , |z1 |22 eiθ )) = P where z 2 ∈ C\{0}. 1 It is not hard to see κ1 (z 1 , 0), (0, z 2 )) = 0 in (2.25). On the other hand, for any  (z, w) = λ1 (z 1 , z 1 ) + λ2

z2 z2 z2 , 1 2 |z 1 |







z2 z2 ∈ span (z 1 , z 1 ) , z 2 , 1 2 |z 1 |

 ,

(2.30)

On the Kähler form of complex L p space and its...

271

where λ1 , λ2 ∈ R, |w|2

 

z 12 z 2 z1 2 z2 = λ1 z 1 + λ2 |z |2 λ 1 z 1 + λ2 |z |2 1 1 = λ21 z 1 z 1 + λ1 λ2 z 1 z 2 + λ2 λ1 z 1 z 2 + λ22 z 2 z 2 = (λ1 z 1 + λ2 z 2 ) (λ1 z 1 + λ2 z 2 ) = |z|2 ,

(2.31)

that implies | wz | = 1. Therefore we have 

 κ(z,w)

z2 z2 (z 1 , z 1 ) , z 2 , 1 2 |z 1 |



  z 12 z 2 3 z1 I m (z 2 z 1 ) + I m 2 |z 1 |2    z w z 12 z 2 1 z1 − z1 z2 − Im | | 2 w z |z 1 |2

3 = 2



3 (I m (z 2 z 1 ) + I m (z 1 z 2 )) 2 = 0.

=

(2.32)

z2 z

So κ vanishes on span((z 1 , z 1 ), (z 2 , |z1 |22 )) for any z 1 , z 2 ∈ C\{0} , I m( zz11 ) = 0. 1 Conversely, suppose that κ vanishes on a plane P spanned by (z 1 , w1 ) and (z 2 , w2 ). From (2.25), we know that 

   1 w 1 z 1 + | | I m (z 2 z 1 ) + 1 + | | I m (w2 w1 ) 2 z 2 w   1 z w + Im | | (w2 z 1 − w1 z 2 ) = 0 2 w z

(2.33)

holds for any (z, w) ∈ span((z 1 , w1 ), (z 2 , w2 )). In the following argument, we divide it into three cases to discuss in terms of | wz | and wz | wz |. The first case is that | wz | = λ for some fixed λ > 0. let (z, w) = λ1 (z 1 , w1 ) + λ2 (z 2 , w2 ) for any λ1 , λ2 ∈ R, then |λ1 w1 + λ2 w2 | = λ|λ1 z 1 + λ2 z 2 |, that implies |w1 | = λ|z 1 |, |w2 | = λ|z 2 | and Re(w1 w2 ) = λ2 Re(z 1 z 2 ). It follows that either z2 z

w1 = λeiθ z 1 , w2 = λeiθ z 2 , or w1 = λeiθ z 1 , w2 = λeiθ |z1 |22 for some θ ∈ [0, 2π ). 1 In the sub-case of w1 = λeiθ z 1 , w2 = λeiθ z 2 for some θ ∈ [0, 2π ), by (2.33) we have     1 λ I m (z 2 z 1 ) + 1 + 1+ λ2 I m (z 2 z 1 ) + λ I m (z 2 z 1 ) 2 2λ = (1 + λ)2 I m (z 2 z 1 ) = 0, (2.34) which implies I m(z 2 z 1 ) = 0 and furthermore I m(w2 w1 ) = 0. That means (z 1 , w1 ) and (z 2 , w2 ) are colinear. So this case can’t occur.

272

Y. Liu z2 z

However, for the other sub-case of w1 = λeiθ z 1 , w2 = λeiθ |z1 |22 for some θ ∈ 1 [0, 2π ), by (2.33) we have    

1 λ 1+ I m (z 2 z 1 ) + 1 + λ2 I m (z 1 z 2 ) = 1 − λ2 I m (z 2 z 1 ) = 0. (2.35) 2 2λ Then λ = 1 or I m(z 2 z 1 ) = 0, but (z 1 , w1 ) and (z 2 , w2 ) can not be colinear. So we have λ = 1 which gives  P = span



z1, z1e

iθ





z2 z2 , z 2 , 1 2 eiθ |z 1 |

 ,

(2.36)

where z 1 , z 2 ∈ C\ {0} and I m(z 1 z 2 ) = 0 for some θ ∈ [0, 2π ) , that finishes the first case. The second case is wz | wz | = eiθ for some fixed θ ∈ [0, 2π ). Let w1 = λ1 eiθ z 1 , w2 = λ2 eiθ z 2 for some λ1 , λ2 > 0. Then it follows from (2.33) that     1 1 λ1 I m (z 2 z 1 ) + 1 + 1+ λ1 λ2 I m(z 2 z 1 ) + (λ1 + λ2 )I m(z 2 z 1 ) 2 2λ1 2     1 1 λ2 I m (z 2 z 1 ) + 1 + = 1+ λ1 λ2 I m(z 2 z 1 ) + (λ1 + λ2 ) I m (z 2 z 1 ) 2 2λ2 2 = (1 + λ1 ) (1 + λ2 ) I m (z 2 z 1 ) =0 (2.37) at the points (z 1 , w1 ) and (z 2 , w2 ), which implies I m(z 2 z 1 ) = 0 and furthermore I m(w2 w1 ) = 0. Thus z 1 and z 2 , w1 and w2 , are colinear, which implies that P equals a plane spanned by one vector from {(z 1 , 0), (z 2 , 0)} and the other from {(0, w1 ), (0, w2 )}. Thus P ∈ T2 . The last case is the negative to the first one and the second one. It gives I m(z 2 z 1 ) = I m(w2 w1 ) = 0 and w2 z 1 − w1 z 2 = 0 in (2.25) because of the linear independence, but the former implies the latter by linear transformation, so it is brought down to I m(z 2 z 1 ) = I m(w2 w1 ) = 0. Thus we have P ∈ T2 by the second case, and that concludes the proof.  

3 Lagrangians of L p space In this section , we consider general p , 1 ≤ p < ∞ but p = 2. For 1 ≤ p < ∞ but p = 2, we have the following theorem: Theorem 3.1 The set of Lagrangian subspaces of C2 with complex L p norm, 1 ≤ p < ∞ but p = 2, is where T2 := {span((z, 0), (0, w)) : z, w ∈ U (1)} ∼ = U (1) × U (1)

(3.1)

On the Kähler form of complex L p space and its...

273

and T1 := {P : P = {λ(z, w) : λ ∈ R, z, w ∈U (1), zw is a constant in U (1)}} ∼ = U (1). (3.2) Proof The proof for this claim is a generalization of the one for 2.6. As we showed in the first part of the proof to 2.6, P = {λ (z, w) : λ ∈ R, z, w ∈ U (1) , zw is a constant in U (1)}

(3.3)

is identical to some 

P := span



z1, z1e

iθ





z2 z2 , z 2 , 1 2 eiθ |z 1 |

 (3.4)

where z 1 , z 2 ∈ C\{0} . Suppose κ p vanishes on a plane P spanned by (z 1 , w1 ) and (z 2 , w2 ). It follows from (2.12) that 

 z 2p −2 p w p p z

| | I m (z 2 z 1 ) + 1 + | | p 1 + | | p I m (w2 w1 ) 2 z w 2 w   2 −2



p w p w z 1 + | |p + 1− | |p I m (3.5) (w2 z 1 − w1 z 2 ) = 0 2 z z w

1+|

w p | z

 2 −2  p

1+

for any (z, w) ∈ span((z 1 , w1 ), (z 2 , w2 )). Analogous to 2.6, we have three cases to consider. The first case is that | wz | = λ for some fixed λ > 0. let (z, w) = λ1 (z 1 , w1 ) + λ2 (z 2 , w2 ) for any λ1 , λ2 ∈ R, then |λ1 w1 + λ2 w2 | = λ|λ1 z 1 + λ2 z 2 |, that implies |w1 | = λ|z 1 |, |w2 | = λ|z 2 | and Re(w1 w2 ) = λ2 Re(z 1 z 2 ). It follows that either z2 z

w1 = λeiθ z 1 , w2 = λeiθ z 2 , or w1 = λeiθ z 1 , w2 = λeiθ |z1 |22 for some θ ∈ [0, 2π ). 1

In the sub-case of w1 = λeiθ z 1 , w2 = λeiθ z 2 for some θ ∈ [0, 2π ), by (3.5) we have 

2    p 1 p p

1 p −2 1+λ 1+ 1 + λ I m (z 2 z 1 ) + 1 + p λ2 I m (z 2 z 1 ) 2 2 λp λ  2 −2 p  1 + λ p p λ p I m (z 2 z 1 ) +2 1 − 2

 2 p p −2 = 1+λ 1 + 2λ p + λ2 p I m (z 2 z 1 ) p

 2 −2 p

  2 −2  2 = 1 + λp p 1 + λ p I m (z 2 z 1 ) = 0,

(3.6)

which gives I m(z 2 z 1 ) = 0 since p ≥ 1, and then I m(w2 w1 ) = 0, that means (z 1 , w1 ) and (z 2 , w2 ) are colinear. So this case is not allowed.

274

Y. Liu z2 z

However, for the other sub-case of w1 = λeiθ z 1 , w2 = λeiθ |z1 |22 for some θ ∈ 1 [0, 2π ), by (3.5) we have 

2    p 1 p p

1 p −2 1+ 1+λ 1 + λ I m (z 2 z 1 ) − 1 + p λ2 I m (z 2 z 1 ) 2 2 λp λ

  2 −2 1 − λ2 p I m (z 2 z 1 ) = 1 + λp p p

 2 −2 p

= 0,

(3.7)

Then λ = 1 or I m (z 2 z 1 ) = 0, but (z 1 , w1 ) and (z 2 , w2 ) can not be colinear. So we have λ = 1 which gives  P = span



z 1 , z 1 eiθ





z2 z2 , z 2 , 1 2 eiθ |z 1 |

 ,

(3.8)

where z 1 , z 2 ∈ C\{0} and I m(z 1 z 2 ) = 0 for some θ ∈ [0, 2π ) , that finishes the first case. The second case is wz | wz | = eiθ for some fixed θ ∈ [0, 2π ). Let w1 = λ1 eiθ z 1 , w2 = λ2 eiθ z 2 for some λ1 , λ2 > 0. Then it follows from (3.5) that   2 −2    p p

1 p p 1 p  2p −2 1 + λ1 I m (z 2 z 1 ) + 1 + p 1 + λ1 1+ λ1 λ2 I m (z 2 z 1 ) 2 2 λ1p λ1 p  p  2 −2 p−1 1 + λ1 p λ1 (λ1 + λ2 ) I m (z 2 z 1 ) + 1− 2

 p  2p −2 p p−1 2 p−1 1 + λ1 + λ1 λ2 + λ1 = 1 + λ1 λ2 I m (z 2 z 1 )

 p  2 −1 p−1 1 + λ1 λ2 I m (z 2 z 1 ) = 1 + λ1 p =0 and

(3.9)

 p  2 −1 2 p−1 1 + λ2 1 + λ2 p λ1 I m (z 2 z 1 ) = 0

(3.10)

at the points (z 1 , w1 ) and (z 2 , w2 ), which implies I m (z 2 z 1 ) = 0 and furthermore I m (w2 w1 ) = 0. Thus z 1 and z 2 , w1 and w2 , are colinear, which implies that P equals a plane spanned by one vector from {(z 1 , 0) , (z 2 , 0)} and the other from {(0, w1 ) , (0, w2 )}. Thus P ∈ T2 . The last case is the negative to the first one and the second one. It gives I m(z 2 z 1 ) = I m(w2 w1 ) = 0 and w2 z 1 − w1 z 2 = 0 in (3.5) because of the linear independence, but the former implies the latter by linear transformation, so it is brought down to I m(z 2 z 1 ) = I m(w2 w1 ) = 0. Thus we have P ∈ T2 from the second case. Thus the claim follows.  

On the Kähler form of complex L p space and its...

275

Remark 3.2 Comparing the results from 3.1 and 2.4, we know (T2 ∪ T1 ) ⊂  T since I m(λ2 z 2 z 1 ) = I m(λ2 w1 w2 )

(3.11)

if z 1 w1 = z 2 w2 , where z 1 , w1 , z 2 , w2 ∈ U (1), so the set of Lagrangian subspaces of C2 with complex L 2 norm is the largest among complex L p norms, 1 ≤ p < ∞. Furthermore, let’s give an example, in which there is a Lagrangian plane for L 2 but not for L p , 1 ≤ p < ∞, p = 2. Example 3.3 Let’s take (z 1 , w1 ) = (2, i) and (z 2 , w2 ) = (i, 2). Since I m(z 2 z 1 ) + I m(w2 w1 ) = 2i − 2i = 0

(3.12)

in (2.21), then the plane span ((2, i) , (i, 2)) is a Lagrangian plane for L 2 . However, plugging them in 2    p w p w p p −2 1 + | | I m (z 2 z 1 ) κp = 1 + | | z 2 z 2

p z

z p −2 1 + | | p I m (w2 w1 ) + 1 + | |p w  2 w z

p

w p 2p −2 w p + 1− 1+| | | | Im (w2 z 1 − w1 z 2 ) , 2 z z w

(3.13)

we have that   

 2 −2  p p 1 1 + · 2 p · 2 + 22 p−2 1 + · p κp = 1 + 2p p 2 2 2  p p 3 ·2 · · (−2) + 1 − 2 4     2   p 3 p −2 · 2 p − 22 p−2 1 − · 2 p−2 + = 2 1 + 2p p + 2 4 8

2   −2 1 + 3 · 2 p−2 − 22 p−2 = 2 1 + 2p p

 2 −2  1 + 3 · 2 p−2 − 22 p−2 = 2 1 + 2p p

(3.14)

at the point (i, 2). Now let

then

f ( p) = 1 + 3 · 2 p−2 − 22 p−2 ,

(3.15)

f ( p) = ln 2 · 2 p−2 (3 − 2 p+1 ) < 0.

(3.16)

Hence f ( p) is strictly decreasing. But f (1) = 23 and f (2) = 0, and so p = 2 is the only zero of f ( p) for 1 ≤ p < ∞. Thus, by (3.14), we conclude that κ p = 0 only if p = 2.

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Remark 3.4 We have now analyzed the differential structure of the complex L p space, by considering its Kähler form. Kähler form, as we know, is a symplectic form, but the symplectic structure of tangent spaces of complex L p space gives the symplectic structure of the space of geodesics in the complex Minkowski space, and in general, the measures on a space or manifold in integral geometry depend on the differential structures on the space or manifold. Holmes–Thompson volumes are defined based on symplectic structure (see, for instance, [2] and [16]), so, as an application, the symplectic structure determined by the Kähler form can be used to determine the symplectic form of the complex Holmes–Thompson volumes restricted on complex lines in integral geometry of complex L p space, about or related to which one can see, for instance, [3], and [15,21]. In a more general setting, the theory of integral geometry on complex Finsler space, by using complex L p space as a model or template, may be established, because Finsler geometry is Riemannian geometry without the quadratic restriction (see [4]), and the classic result in integral geometry, Crofton formula, still holds in projective Finsler space, on which one can see, for instance, [18] and [22]. Acknowledgments The author would like to thank Prof. J. Fu for some helpful discussions in this subject. This work was partially supported by NSF.

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