## 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

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. [2–4]), 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,

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

## 2. Gelfand transform

Given a double fibration:

where

which is a 1-density

## 3. 1-Density

**Lemma 3.1**. For any

*Proof*. For

such that

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

Thus, we have now proved the Crofton formula: Given a differentiable curve

## 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*

*and*

*Proof*. First, we can show that

is identical to some

where

We can get

where

that implies

So

Conversely, suppose that

holds for any

The first case is that

In the sub-case of

which implies

However, for the other sub-case of

Then

where

The second case is

at the points *P* equals a plane spanned by one vector from

The last case is the negative to the first one and the second one. It gives

## 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 *F* is a Finsler metric. The natural symplectic form on *F* is

Define a projection

Let

**Theorem 5.1. π*ω0=ω|S*R2.**

*Proof*. Applying the equality

we obtain

By the positive homogeneity of *F*, one can get the useful fact that

By differentiating (24), we get

Applying (22) again, we have

Thus, the claim follows.

Remark 5.2. For a *n*-dimensional Minkowski space

Therefore, letting *F* be a Finsler metric on *S*_{F} be the unit sphere in the Minkowski space

**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

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. [21–23].

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

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

As for Minkowski plane, it is a normed two dimensional space with a norm *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 *S*^{1} such that

if *F* is an even *C*^{4} function on *S*^{1}. 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,

where

*Proof*. First, consider the case of

Thus, we have

by using the classic Crofton formula.

For the general case of *θ* by

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

for any curve

for Minkowski plane.

The Holmes-Thompson area *U* in a Minkowski plane is defined as *ω*_{0} is the natural symplectic form on the cotangent bundle of *ω* to the space of affine lines *ω*.

## 7. Application to HT area and related

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

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

starting at

we can obtain a fact that

where *F*, for all

Let *p*′ is the projection (composition) from

Apply the above fact and

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

Furthermore, combining with (33), we have

where

To obtain the HT area, one can define a map

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*

*Proof*. On the one hand,

On the other hand,

where

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.