Open access peer-reviewed chapter

Lagrangian Subspaces of Manifolds

By Yang Liu

Submitted: May 12th 2016Reviewed: December 16th 2016Published: May 3rd 2017

DOI: 10.5772/67290

Downloaded: 738

Abstract

In this chapter, we provide an overview on the Lagrangian subspaces of manifolds, including but not limited to, linear vector spaces, Riemannian manifolds, Finsler manifolds, and so on. There are also some new results developed in this chapter, such as finding the Lagrangians of complex spaces and providing new insights on the formula for measuring length, area, and volume in integral geometry. 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 Finsler space. Moreover, we show that the space of oriented lines and the tangent bundle of unit sphere in Minkowski space are symplectomorphic.

Keywords

  • Lagrangian subspace
  • differential geometry

1. Introduction

In differential geometry and differential topology, manifolds are the main objects being studied, and Lagrangian submanifolds are submanifolds that carry differential forms with special property, which are usually called symplectic form in real manifolds and Kahler form in complex manifolds.

This book chapter is concerned with explicit canonical symplectic form for real and complex spaces and answer to the questions on the existence of Lagrangian subspace. One can find and explicitly describe the set of Lagrangian subspaces of R2with Lpnorm, 1p<, as a an example of Finsler spaces. Since Holmes-Thompson volumes, as measures, depend on the differential structures of the spaces, the symplectic structure determined by the symplectic form can be used to determine the symplectic form of Holmes-Thompson volumes restricted on lines in integral geometry of Lpspaces, as an application to integral geometry.

Some ingenuous ideas in physics and engineering actually originated from mathematics. For example, the relativity theory in physics, to some sense, originated from Riemmanian geometry. The real Finsler spaces, as generalizations of real Riemannian manifolds, were introduced in Ref. [1] about a century ago and have been studied by many researchers (see, for instance, Refs. [24]), and Finsler spaces (see, for instance, Refs. [5, 6]) have become an interest of research for the studies of geometry, including differential geometry and integral geometry, in recent decades. By the way, there are applications of Finsler geometry in physics and engineering, and in particular, Finsler geometry can be applied to engineering dynamical systems, on which one can see Ref. [7]. As a typical Finsler space, Lpspace, 1<p<, has the main features of a Finsler space. As such, we focus on Lpspace, 1<p<, in this chapter, but some results can be generalized to general Finsler spaces, on which one can refer to Ref. [8]. The Lpspace, 1<p<, as a generalization of Euclidean space, has a rich structure in functional analysis (see, for instance, Refs. [9, 10]), and particularly in Banach space. Furthermore, it has broad applications in statistics (see, for instance, Refs. [11, 12]), engineering (see, for instance, Ref. [13, 27]), mechanics (see, for instance, Ref. [14]), computational science (see, for instance, Ref. [15]), biology (see, for instance, Ref. [16]), and other areas. Along this direction, Lp, 0<p1, in the sense of conjugacy to the scenario of Lp, 1<p<, also has broad applications, in particular, signal processing in engineering, on which one can see Refs. [1719].

This chapter is structured as follows: In Section 2, we provide a description on Gelfand transform, which is one of the most fundamental transforms in integral geometry; in Section 3, we introduce density needed for the measure of length of curves; in Section 4, we further study the Lagrangian subspaces of complex Lpspaces; in Section 5, we work on tangent bundle of unit sphere in Minkowski space and its symplectic or Lagrangian structure; in Section 6, we apply the Lagrangian structure to establish the length formula in integral geometry; and in Section 7, we further apply the Lagrangian structure of a Minkowski space to establish the formula for the Holmes-Thompson area in integral geometry.

2. Gelfand transform

Given a double fibration:

R2π1Fπ2Gr1(R2)¯E1

where

F={((x,y),l(r,θ)):(x,y)R2,l(r,θ)Gr1(R2)¯,(x,y)l(r,θ)}{(x,y,r,θ):xcos(θ)+ysin(θ)=r},

π1and π2are the natural projections of fibers. The Gelfand transform of a 2-density φ=|drdθ|is defined as

GT(φ)=π1*π2*φ,E2

which is a 1-density R2.

3. 1-Density

Lemma 3.1. For any v=(α,β)T(x,y)R2,

GT(φ)((x,y),v)=4|v|.E3

Proof. For v=(α,β)T(x,y)R2,there exists

v˜=(α,β,αcos(θ)+βsin(θ),θ)T((x,y),l(r,θ))F,E4

such that dπ1(v˜)=v. Therefore, we have

GT(φ)(v)=π11((x,y))π2*φ(v˜,)={(x,y),l(p,θ):xcos(θ)+ysin(θ)=r}|drdθ|(v˜,)=02π|αcos(θ)+βsin(θ)|dθ=02π|v(cos(θ),sin(θ))|dθ=|v|02πcos(θ0+θ)dθwhereα=|v|cos(θ0),β=|v|sin(θ0)=4|v|.E5

Remark 3.2. By Alvarez’s Gelfand transform for Crofton type formulas, we know that

lR2#(γl(r,θ))drdθ=γGT(φ).E6

Thus, we have now proved the Crofton formula: Given a differentiable curve γin R2, the length of γcan be computed in the following formula:

Length(γ)=14lR2#(γl(r,θ))drdθ.E7

4. Lagrangian subspaces of complex spaces

Some of the results have obtained in Ref. [8], but because the Lagrangian subspaces of complex spaces are essential to establish the generalized volume formula in complex integral geometry, let us give an expository on the Kahler strut rue of generalized complex spaces.

Theorem 4.1. The set of Lagrangian subspaces of C2 with L1 norm is T2∪T1, where

T2:={span((z,0),(0,w)):z,wU(1)}U(1)×U(1)E8

and

T1:={P:P={λ(z,w):λR,z,wU(1),zwisaconstantinU(1)}}U(1).E9

Proof. First, we can show that

P={λ(z,w):λR,z,wU(1),zwisaconstantinU(1)}E10

is identical to some

P:=span((z1,z1eiθ),(z2,z12z2¯|z1|2eiθ))E11

where z1,z2C\{0}. For any λ(eiφ,eiψ)P, let z1=λeiφ, θ=ψφ, we have P=span((z1,z1eiθ),(z2,z12z2¯|z1|2eiθ))=Pwhere z2C\{0}.

We can get κ1(z1,0),(0,z2))=0. On the other hand, for any

(z,w)=λ1(z1,z1)+λ2(z2,z12z2¯|z1|2)span((z1,z1),(z2,z12z2¯|z1|2)),E12

where λ1,λ2R,

|w|2=(λ1z1+λ2z12z2¯|z1|2)(λ1z1¯+λ2z1¯2z2|z1|2)=λ12z1z1¯+λ1λ2z1¯z2+λ2λ1z1z2¯+λ22z2¯z2=(λ1z1+λ2z2)(λ1z1¯+λ2z2¯)=|z|2,E13

that implies |wz|=1. Therefore, we have

κ(z,w)((z1,z1),(z2,z12z2¯|z1|2))=32(Im(z2z1¯)+32Im(z12z2¯|z1|2z1¯))12Im(zw|wz|(z12z2¯|z1|2z1¯z1z2¯))=32(Im(z2z1¯)+Im(z1z2¯))=0.E14

So κvanishes on span((z1,z1),(z2,z12z2¯|z1|2))for any z1,z2C\{0}, Im(z1z1)0.

Conversely, suppose that κvanishes on a plane Pspanned by (z1,w1)and (z2,w2). We know that

(1+12|wz|)Im(z2z1¯)+(1+12|zw|)Im(w2w1¯)+12Im(zw|wz|(w2z1¯w1z2¯))=0E15

holds for any (z,w)span((z1,w1),(z2,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|=λfor some fixed λ>0. Let (z,w)=λ1(z1,w1)+λ2(z2,w2)for any λ1,λ2R, then |λ1w1+λ2w2|=λ|λ1z1+λ2z2|, that implies |w1|=λ|z1|, |w2|=λ|z2|and Re(w1w2¯)=λ2Re(z1z2¯). It follows that w1=λeiθz1, w2=λeiθz2, or w1=λeiθz1, w2=λeiθz12z2¯|z1|2for some θ[0,2π).

In the sub-case of w1=λeiθz1, w2=λeiθz2for some θ[0,2π), by Eq. (15) we have

(1+λ2)Im(z2z1¯)+(1+12λ)λ2Im(z2z1¯)+λIm(z2z1¯)=(1+λ)2Im(z2z1¯)=0,E16

which implies Im(z2z1¯)=0and furthermore Im(w2w1¯)=0. That means (z1,w1)and (z2,w2)are colinear. So this case cannot occur.

However, for the other sub-case of w1=λeiθz1, w2=λeiθz12z2¯|z1|2for some θ[0,2π), by Eq. (15) we have

(1+λ2)Im(z2z1¯)+(1+12λ)λ2Im(z1z2¯)=(1λ2)Im(z2z1¯)=0.E17

Then λ=1or Im(z2z1¯)=0, but (z1,w1)and (z2,w2)cannot be colinear. So, we have λ=1which gives

P=span((z1,z1eiθ),(z2,z12z2¯|z1|2eiθ)),E18

where z1,z2C\{0}and Im(z1z2¯)0for some θ[0,2π). This finishes the first case.

The second case is wz|wz|=eiθfor some fixed θ[0,2π). Let w1=λ1eiθz1,w2=λ2eiθz2for some λ1,λ2>0. Then it follows from (15) that

(1+λ12)Im(z2z1¯)+(1+12λ1)λ1λ2Im(z2z1¯)+12(λ1+λ2)Im(z2z1¯)=(1+λ22)Im(z2z1¯)+(1+12λ2)λ1λ2Im(z2z1¯)+12(λ1+λ2)Im(z2z1¯)=(1+λ1)(1+λ2)Im(z2z1¯)=0E19

at the points (z1,w1)and (z2,w2), which implies Im(z2z1¯)=0and furthermore Im(w2w1¯)=0. Thus, z1and z2, w1, and w2are colinear, which implies that P equals a plane spanned by one vector from {(z1,0),(z2,0)}and the other from {(0,w1),(0,w2)}. Thus PT2.

The last case is the negative to the first one and the second one. It gives Im(z2z1¯)=Im(w2w1¯)=0and w2z1¯w1z2¯=0because of the linear independence, but the former implies the latter by linear transformation, so it is brought down to Im(z2z1¯)=Im(w2w1¯)=0. Thus, we have PT2by the second case, and that concludes the proof.

5. Tangent bundle of uni-sphere in Minkowski space and symplectic or Lagrangian structure

In this section, we show that the space of oriented lines and the tangent bundle of unit sphere in Minkowski space are symplectomorphic.

Let us consider a Minkowski plane (R2,F)first, where F is a Finsler metric. The natural symplectic form on T*R2is dxdξ¯+dydη¯, and then the natural symplectic form on TR2induce by the Finsler metric F is

ω:=dxdFξ+dydFη=2Fξ2dxdξ+2Fξη(dxdη+dydξ)+2Fη2dydη.E20

Define a projection π:TR2Gr1(R2)¯by

π((x,y);(ξ,η))=((x,y)dF(ξ,η)((x,y))(ξ,η);(ξ,η)).E21

Let SFbe the unit circle in the Minkowski plane and TSFbe its tangent bundle. It is a fact that TSFGr1(R2)¯. On the other hand, since TSFis embedded in TR2, it inherits a natural symplectic form ω0:=ω|TSFfrom TR2.

Theorem 5.1. π*ω0=ω|S*R2.

Proof. Applying the equality

Fξdξ+Fηdη=0,E22

we obtain

π*ω0=2Fξ2d(xdF(ξ,η)((x,y))ξ)dξ+2Fξη(d(xdF(ξ,η)((x,y))ξ)dη+d(ydF(ξ,η)((x,y))η)dξ)+2Fη2d(ydF(ξ,η)((x,y))η)dη=2Fξ2dxdξ+2Fξη(dxdη+dydξ)+2Fη2dydηd(dF(ξ,η)((x,y)))(2Fξ2ξdξ+2Fη2ηdη+2Fξη(ξdη+ηdξ)).E23

By the positive homogeneity of F, one can get the useful fact that F(ξ,η)=ξFξ+ηFη. Therefore,

ξFξ+ηFη=1.E24

By differentiating (24), we get

2Fξ2ξdξ+2Fη2ηdη+2Fξη(ξdη+ηdξ)+Fξdξ+Fηdη=0.E25

Applying (22) again, we have

2Fξ2ξdξ+2Fη2ηdη+2Fξη(ξdη+ηdξ)=0.E26

Thus, the claim follows.

Remark 5.2. For a n-dimensional Minkowski space (Rn,F), we just need to add more indices, then the theorem above is also true for (Rn,F).

Therefore, letting F be a Finsler metric on Rnand SF be the unit sphere in the Minkowski space (Rn,F), we obtain the following general theorem:

Theorem 5.3. The symplectic form on the space of lines in a Minkowski space (Rn,F) is the canonical symplectic form on the tangent bundle TSF as imbedded in TRn.

We have the following remarks:

Remark 5.4. Theorem 5.3 provides a perspective that we can transform calculus on Gr1(R2)¯to ones on TSF.

and

Remark 5.5. We can analyze the differential structure of the Minkowski space by considering its symplectic form or Lagrangian structure. The Lagrangian structure of tangent spaces of Minkowski space gives the symplectic structure of the space of geodesics in the Minkowski space, and in general, the measures on a space or manifold in integral geometry depend on the differential structures of the space or manifold. Holmes-Thompson volumes are defined based on Lagrangian structure (see, for instance, Refs. [12, 20]), so, as an application, the symplectic structure determined by the symplectic form can be used to determine the symplectic form of the Holmes-Thompson volumes restricted on lines in integral geometry of Minkowski space, about which one can see Refs. [2123].

Another remark from the proof of Theorem 5.1 is that

Remark 5.6. A combination of (26) and Gelfand transform (see Ref. [6]) may be used to provide a short proof of the general Crofton formula for Minkowski space.

6. Application to generalized length and related

For any rectifiable curve γ in the Euclidean plane, the classic Crofton formula is

Length(γ)=14002π#(γl(r,θ))dθdr,E27

where θ is the angle from the x-axis to the normal of the oriented line l and r is the distance form the origin to l. Let us denote the affine l-Grassmannians consisting of lines in R2by Gr1(R2)¯.

As for Minkowski plane, it is a normed two dimensional space with a norm F()=||||, in which the unit disk is convex and F has some smoothness.

Two significant and useful tools that are used to obtain the Crofton formula for Minkowski plane are the cosine transform and Gelfand transform. Let us explain them one by one first and see the connections between them later. A important fact or result from spherical harmonics about cosine transform is that there is some even function on S1 such that

F()=S1|ξ,|g(θ)dθ,E28

if F is an even C4 function on S1. A great reference for this would be [24] by Groemer. As for Gelfand transform, it is the transform of differential forms and densities on double fibrations, for instance, R2π1Iπ2Gr1(R2)¯, where I:={(x,l)R2×Gr1(R2)¯:xl}is the incidence relations and π1and π2are projections. A formula one can take as an example of the fundamental theorem of Gelfand transform is the following:

γπ1*π2*|Ω|=lGr1(R2)¯#(γl)|Ω|,E29

where Ω:=g(θ)dθdr. However, here we provide a direct proof for this fundamental theorem of Gelfand transform.

Proof. First, consider the case of Ω=dθdr. For any vTxγ, since there is some vTxI, such that π1*(v)=v, then

(π1*π2*|Ω|)x(v)=(π11(x)π2*|Ω|)x(v)=xπ11(x)(π2*|Ω|)x(v)=S1(π2*|dθdr|)(v)=S1|dr(π2*(v))|dθ=S1|v,θ|dθ=4|v|.E30

Thus, we have

γπ1*π2*|Ω|=4Length(γ)=lGr1(R2)¯#(γl)|Ω|E31

by using the classic Crofton formula.

For the general case of Ω=f(θ)dθdr, we just need to substitute dθ by g(θ)dθin the equalities in the first case.

Furthermore, we can also see, from the above proof and eq:exist, that

γπ1*π2*|Ω|=ab(π1*π2*|Ω|)(γ(t))dt=ab4F(γ(t))dt=4Length(γ),E32

for any curve γ(t):[a,b]R2differentiable almost everywhere in the Minkowski space. Therefore, by using (29), we obtain that

Length(γ)=14lGr1(R2)¯#(γl)|g(θ)dθdr|E33

for Minkowski plane.

The Holmes-Thompson area HT2(U)of a measurable set U in a Minkowski plane is defined as HT2(U):=1πD*U|ω0|2, where ω0 is the natural symplectic form on the cotangent bundle of R2and D*U:={(x,ξ)T*R2:F*(ξ)1}. To study it from the perspective of integral geometry, we need to introduce a symplectic form ω to the space of affine lines Gr1(R2)¯and construct an invariant measure based on ω.

7. Application to HT area and related

Now let us see the Crofton formula for Minkowski plane, which is

Length(γ)=14Gr1(R2)¯#(γl)|ω|.E34

To prove this, it is sufficient to show that it holds for any straight line segment

L:[0,||p2p2||]R2,L(t)=p1+p2p1||p2p1||t,E35

starting at p1and ending at p2in R2. First, using the diffeomorphism between the circle bundle and co-circle bundle, which is

φF:SR2S*R2φF(x,ξ)=(x,dFξ),E36

we can obtain a fact that

L×{p2p1p2p1}φF*α0=φF(L×{p2p1p2p1})α0=0||p2p1||α0dFp2p1||p2p1||((p2p1||p2p1||,0))dt=0||p2p1||dFp2p1||p2p1||(p2p1||p2p1||)dt,E37

where α0is the tautological one-form, precisely α0ξ(X):=ξ(π0*X)for any XTξT*R2, and dα0=ω0. Applying the basic equality that dFξ(ξ)=1, which is derived from the positive homogeneity of F, for all ξSR2, the above quantity becomes 0||p2p1||1dt, which equals ||p2p1||.

Let R:={ξxS*R2:xp1p2¯}and T={lGr1(R2)¯:lp1p2¯Ø}, and p′ is the projection (composition) from S*R2to Gr1(R2)¯.

Apply the above fact and p*ω=ω0,

T|ω|=p(R)|ω|=R|p*ω|=R|ω0|=|R+ω0|+|Rω0|=|R+α0|+|Rα0|=4||p2p1||.E38

Thus, we have shown the Crofton formula for Minkowski plane.

Furthermore, combining with (33), we have

14lGr1(R2)¯#(γl)|Ω|=14Gr1(R2)¯#(γl)|ω|,E39

where Ω=g(θ)dθdr. Then, by the injectivity of cosine transform in Ref. [24], |Ω|=|ω|.

To obtain the HT area, one can define a map

π:Gr1(R2)¯×Gr1(R2)¯\ΔR2π(l,l)=ll,E40

extended from Alvarez’s construction of taking intersections. The following theorem can be obtained.

Theorem 7.1. For any bounded measurable subset U of a Minkowski plane, we have

HT2(U)=12πxR2χ(xU)|π*Ω2|.E41

Proof. On the one hand,

1πD*Uω02=1πD*Uω02=1πS*Uα0ω0.E42

On the other hand,

1πxR2χ(xU)π*Ω2=1π{(l,l)Gr1(R2)¯×Gr1(R2)¯\Δ:llU}Ω2=1π{(l,l)Gr1(R2)¯×Gr1(R2)¯\Δ:llU}ω2=1πT*U\{(x,ξ,ξ):ξSx*U}p*ω2=1πT*U\{(x,ξ,ξ):ξSx*U}ω02=2π{(x,ξ,ξ):ξSx*U}α0ω0=2πS*Uα0ω0,E43

where

T*U:={(x,ξ,ξ):ξ,ξSx*U}.E44

So the claim follows.

Remark 7.2. Lagrangian structure provides the underlying differential structure needed to measure the Holme-Thompson area in integral geometry and therefore is essential and doundamental in integral geometry. For Finsler manifolds, real or complex, it is necessary to analyze the Lagrangian structure of the Finsler manifolds, in the forms of symplectic structure and Kahler structure, and many Finsler manifolds may not have a Lagrangian structure, about which one can refer to Ref. [25]. However, for smooth projective Finsler spaces, the integral geometry formulas have been studied in Ref. [26], for instance.

Acknowledgments

The author would like to thank his family for their constant support for his academic career since his doctoral study in the USA, for the partial support by the National Science Foundation, and for the partial support by Air Force Office of Scientific Research under Grant AFOSR 9550-12-1-0455, and the author would also like to give thanks to Dr. P. Dang. Besides, the author would like to thank the reviewer for his or her helpful comments.

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Yang Liu (May 3rd 2017). Lagrangian Subspaces of Manifolds, Lagrangian Mechanics, Hüseyin Canbolat, IntechOpen, DOI: 10.5772/67290. Available from:

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