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Engineering » Mechanical Engineering » "Lagrangian Mechanics", book edited by Hüseyin Canbolat, ISBN 978-953-51-3132-8, Print ISBN 978-953-51-3131-1, Published: May 3, 2017 under CC BY 3.0 license. © The Author(s).

# Lagrangian Subspaces of Manifolds

By Yang Liu
DOI: 10.5772/67290

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# Lagrangian Subspaces of Manifolds

Yang Liu1, 2
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## 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 R2 with Lp norm, 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 Lp spaces, 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, Lp space, 1<p<, has the main features of a Finsler space. As such, we focus on Lp space, 1<p<, in this chapter, but some results can be generalized to general Finsler spaces, on which one can refer to Ref. [8]. The Lp space, 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 Lp spaces; 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:

 R2←π1F→π2Gr1(R2)¯ (1)

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}, ()

π1 and π2 are the natural projections of fibers. The Gelfand transform of a 2-density φ=|drdθ| is defined as

 GT(φ)=π1*π2*φ, (2)

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|. (3)

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, (4)

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

 GT(φ)(v)=∫π1−1((x,y))π2*φ(v˜,▪)=∫{(x,y),l(p,θ): xcos(θ)+ysin(θ)=r}|dr∧dθ|(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|. (5)

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

 ∫l∈R2#(γ∩l(r,θ))drdθ=∫γGT(φ). (6)

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

 Length(γ)=14∫l∈R2#(γ∩l(r,θ))drdθ. (7)

## 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 T2T1, where

 T2:={span((z,0),(0,w)):z,w∈U(1)}≅U(1)×U(1) (8)

and

 T1:={P:P={λ(z,w):λ∈R,z,w∈U(1), zw isaconstantin U(1)}}≅U(1). (9)

Proof. First, we can show that

 P={λ(z,w):λ∈R,z,w∈U(1), zw isaconstantin U(1)} (10)

is identical to some

 P′:=span((z1,z1eiθ),(z2,z12z2¯|z1|2eiθ)) (11)

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))=P where 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)), (12)

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, (13)

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. (14)

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

Conversely, suppose that κ vanishes on a plane P spanned by (z1,w1) and (z2,w2). We know that

 (1+12|wz|)Im(z2z1¯)+(1+12|zw|)Im(w2w1¯)+12Im(zw|wz|(w2z1¯−w1z2¯))=0 (15)

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|=λ for 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|2 for some θ[0,2π).

In the sub-case of w1=λeiθz1, w2=λeiθz2 for some θ[0,2π), by Eq. (15) we have

 (1+λ2)Im(z2z1¯)+(1+12λ)λ2Im(z2z1¯)+λIm(z2z1¯)=(1+λ)2Im(z2z1¯)=0, (16)

which implies Im(z2z1¯)=0 and 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|2 for some θ[0,2π), by Eq. (15) we have

 (1+λ2)Im(z2z1¯)+(1+12λ)λ2Im(z1z2¯)=(1−λ2)Im(z2z1¯)=0. (17)

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

 P=span((z1,z1eiθ),(z2,z12z2¯|z1|2eiθ)), (18)

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

The second case is wz|wz|=eiθ for some fixed θ[0,2π). Let w1=λ1eiθz1,w2=λ2eiθz2 for 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¯)=0 (19)

at the points (z1,w1) and (z2,w2), which implies Im(z2z1¯)=0 and furthermore Im(w2w1¯)=0. Thus, z1 and z2, w1, and w2 are 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¯)=0 and w2z1¯w1z2¯=0 because 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 PT2 by 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*R2 is dxdξ¯+dydη¯, and then the natural symplectic form on TR2 induce by the Finsler metric F is

 ω:=dx∧d∂F∂ξ+dy∧d∂F∂η=∂2F∂ξ2dx∧dξ+∂2F∂ξ∂η(dx∧dη+dy∧dξ)+∂2F∂η2dy∧dη. (20)

Define a projection π:TR2Gr1(R2)¯ by

 π((x,y);(ξ,η))=((x,y)−dF(ξ,η)((x,y))(ξ,η);(ξ,η)). (21)

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

Theorem 5.1. π*ω0=ω|S*R2.

Proof. Applying the equality

 ∂F∂ξdξ+∂F∂ηdη=0, (22)

we obtain

 π*ω0=∂2F∂ξ2d(x−dF(ξ,η)((x,y))ξ)∧dξ+∂2F∂ξ∂η(d(x−dF(ξ,η)((x,y))ξ)∧dη +d(y−dF(ξ,η)((x,y))η)∧dξ)+∂2F∂η2d(y−dF(ξ,η)((x,y))η)∧dη=∂2F∂ξ2dx∧dξ+∂2F∂ξ∂η(dx∧dη+dy∧dξ)+∂2F∂η2dy∧dη −d(dF(ξ,η)((x,y)))∧(∂2F∂ξ2ξdξ+∂2F∂η2ηdη+∂2F∂ξ∂η(ξdη+ηdξ)). (23)

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

 ξ∂F∂ξ+η∂F∂η=1. (24)

By differentiating (24), we get

 ∂2F∂ξ2ξdξ+∂2F∂η2ηdη+∂2F∂ξ∂η(ξdη+ηdξ)+∂F∂ξdξ+∂F∂ηdη=0. (25)

Applying (22) again, we have

 ∂2F∂ξ2ξdξ+∂2F∂η2ηdη+∂2F∂ξ∂η(ξdη+ηdξ)=0. (26)

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 Rn and 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)=14∫0∞∫02π#(γ∩l(r,θ))dθdr, (27)

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 R2 by 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, (28)

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 π1 and π2 are projections. A formula one can take as an example of the fundamental theorem of Gelfand transform is the following:

 ∫γπ1*π2*|Ω|=∫l∈Gr1(R2)¯#(γ∩l)|Ω|, (29)

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)=(∫π1−1(x)π2*|Ω|)x(v)=∫x′∈π1−1(x)(π2*|Ω|)x′(v′)=∫S1(π2*|dθ∧dr|)(v′)=∫S1|dr(π2*(v′))|dθ=∫S1|〈v,θ〉|dθ=4|v|. (30)

Thus, we have

 ∫γπ1*π2*|Ω|=4Length(γ)=∫l∈Gr1(R2)¯#(γ∩l)|Ω| (31)

by using the classic Crofton formula.

For the general case of Ω=f(θ)dθdr, we just need to substitute dθ by g(θ)dθ in 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), (32)

for any curve γ(t):[a,b]R2 differentiable almost everywhere in the Minkowski space. Therefore, by using (29), we obtain that

 Length(γ)=14∫l∈Gr1(R2)¯#(γ∩l)|g(θ)dθ∧dr| (33)

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 R2 and 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)=14∫Gr1(R2)¯#(γ∩l)|ω|. (34)

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

 L:[0,||p2−p2||]→R2, L(t)=p1+p2−p1||p2−p1||t, (35)

starting at p1 and ending at p2 in R2. First, using the diffeomorphism between the circle bundle and co-circle bundle, which is

 φF:SR2→S*R2φF(x,ξ)=(x,dFξ), (36)

we can obtain a fact that

 ∫L×{p2−p1‖p2−p1‖}φF*α0=∫φF(L×{p2−p1‖p2−p1‖})α0        =∫0||p2−p1||α0dFp2−p1||p2−p1||((p2−p1||p2−p1||,0))dt        =∫0||p2−p1||dFp2−p1||p2−p1||(p2−p1||p2−p1||)dt, (37)

where α0 is 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*R2 to Gr1(R2)¯.

Apply the above fact and p*ω=ω0,

 ∫T|ω|=∫p′(R)|ω|=∫R|p′*ω|=∫R|ω0|=|∫R+ω0|+|∫R−ω0|=|∫∂R+α0|+|∫∂R−α0|=4||p2−p1||. (38)

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

Furthermore, combining with (33), we have

 14∫l∈Gr1(R2)¯#(γ∩l)|Ω|=14∫Gr1(R2)¯#(γ∩l)|ω|, (39)

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)¯\Δ→R2π(l,l′)=l∩l′, (40)

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)=12π∫x∈R2χ(x∩U)|π*Ω2|. (41)

Proof. On the one hand,

 1π∫D*Uω02=1π∫∂D*Uω02=1π∫S*Uα0∧ω0. (42)

On the other hand,

 1π∫x∈R2χ(x∩U)π*Ω2=1π∫{(l,l′)∈Gr1(R2)¯×Gr1(R2)¯\Δ:l∩l′∈U}Ω2=1π∫{(l,l′)∈Gr1(R2)¯×Gr1(R2)¯\Δ:l∩l′∈U}ω2=1π∫T*U\{(x,ξ,ξ):ξ∈Sx*U}p′*ω2=1π∫T*U\{(x,ξ,ξ):ξ∈Sx*U}ω02=2π∫{(x,ξ,ξ):ξ∈Sx*U}α0∧ω0=2π∫S*Uα0∧ω0, (43)

where

 T*U:={(x,ξ,ξ′):ξ,ξ′∈Sx*U}. (44)

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.

## Acknowledgements

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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