Lagrangian Subspaces of Manifolds

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.

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 ${\mathbb{R}}^2$ with $L^p$ norm, $1\le p<\infty$, 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 $L^p$ 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. [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, $L^p$ space, $1<p<\infty$, has the main features of a Finsler space. As such, we focus on $L^p$ space, $1<p<\infty$, in this chapter, but some results can be generalized to general Finsler spaces, on which one can refer to Ref. [8]. The $L^p$ space, $1<p<\infty$, 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, $L^p$, $0<p\le 1$, in the sense of conjugacy to the scenario of $L^p$, $1<p<\infty$, also has broad applications, in particular, signal processing in engineering, on which one can see Refs. [17–19].

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 $L^p$ 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:

${\mathbb{R}}^2{\displaystyle \stackrel{{\pi }_1}{\leftarrow }}\text{F}{\displaystyle \stackrel{{\pi }_2}{\to }}{\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$E1

where

$\begin{array}{c}\text{F}=\left\{((x\mathrm{,}y),l(r\mathrm{,}\theta )):(x\mathrm{,}y)\in R^2,l(r\mathrm{,}\theta )\in {\displaystyle \overline{G{r}_1(R^2)}},(x\mathrm{,}y)\in l(r\mathrm{,}\theta )\right\}\\ \simeq \left\{(x\mathrm{,}y\mathrm{,}r\mathrm{,}\theta ):x\cos (\theta )+y\sin (\theta )=r\right\}\mathrm{,}\end{array}$

${\pi }_1$ and ${\pi }_2$ are the natural projections of fibers. The Gelfand transform of a 2-density $\phi =|\text{d}r\wedge \text{d}\theta |$ is defined as

$\text{GT}(\phi )={\pi }_{1_{*}}{\pi }_2^{*}\phi \mathrm{,}$E2

which is a 1-density ${\mathbb{R}}^2$.

3. 1-Density

Lemma 3.1. For any $v=(\alpha \mathrm{,}\beta )\in T_{(x\mathrm{,}y)}{\mathbb{R}}^2\mathrm{,}$

$\text{GT}(\phi )((x\mathrm{,}y),v)=4\left|v\right|\mathrm{.}$E3

Proof. For $v=(\alpha \mathrm{,}\beta )\in T_{(x\mathrm{,}y)}{R}^2\mathrm{,}$ there exists

${\displaystyle \tilde{v}}=(\alpha \mathrm{,}\beta \mathrm{,}\alpha \text{cos}(\theta )+\beta \text{sin}(\theta ),\theta )\in T_{((x\mathrm{,}y),l(r\mathrm{,}\theta ))}\text{F}\mathrm{,}$E4

such that $\text{d}{\pi }_1({\displaystyle \tilde{v}})=v$. Therefore, we have

$\begin{array}{ccc}\text{GT}(\phi )(v)\hfill & =\hfill & {\displaystyle {\displaystyle \underset{{\pi }_1^{-1}((x\mathrm{,}y))}{\int }}{\pi }_2^{*}\phi ({\displaystyle \tilde{v}},\blacksquare )}\hfill \\ \hfill & =\hfill & {\displaystyle {\displaystyle \underset{{}_{\left\{(x\mathrm{,}y),l(p\mathrm{,}\theta ):x\text{cos}(\theta )+y\text{sin}(\theta )=r\right\}}}{\int }}|\text{d}r\wedge \text{d}\theta |({\displaystyle \tilde{v}}\mathrm{,}\blacksquare )}\hfill \\ \hfill & =\hfill & {\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}|\alpha \text{cos}(\theta )+\beta \text{sin}(\theta )|d\theta }\hfill \\ \hfill & =\hfill & {\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}|v\cdot (\text{cos}(\theta ),\text{sin}(\theta ))|d\theta }\hfill \\ \hfill & =\hfill & \left|v\right|{\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}\text{cos}({\theta }_0+\theta )\text{d}\theta \text{where}\alpha =\left|v\right|\cos ({\theta }_0),\beta =\left|v\right|\sin ({\theta }_0)}\hfill \\ \hfill & =\hfill & 4\left|v\right|\mathrm{.}\hfill \end{array}$E5

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

${\displaystyle {\int }_{l\in R^2}}\#(\gamma \cap l(r\mathrm{,}\theta ))\text{d}r\text{d}\theta ={\displaystyle {\int }_{\gamma }}\text{GT}(\phi ).$E6

Thus, we have now proved the Crofton formula: Given a differentiable curve $\gamma$ in ${\mathbb{R}}^2$, the length of $\gamma$ can be computed in the following formula:

$\text{Length}(\gamma )=\frac{1}{4}{\displaystyle {\int }_{l\in R^2}}\#(\gamma \cap l(r\mathrm{,}\theta ))\text{d}r\text{d}\theta \mathrm{.}$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 ${\mathbb{C}}^2$ with $L^1$ norm is ${\mathbb{T}}^2\cup {\mathbb{T}}^1$, where

${\mathbb{T}}^2:=\left\{\text{span}((z\mathrm{,0}),(\mathrm{0,}w)):z\mathrm{,}w\in U(1)\right\}\cong U(1)\times U(1)$E8

and

${\mathbb{T}}^1:=\left\{P:P=\left\{\lambda (z\mathrm{,}w):\lambda \in \mathbb{R}\mathrm{,}z\mathrm{,}w\in U(1),zw\text{is}\text{a}\text{constant}\text{in}U(1)\right\}\right\}\cong U(1).$E9

Proof. First, we can show that

$P=\left\{\lambda (z\mathrm{,}w):\lambda \in \mathbb{R}\mathrm{,}z\mathrm{,}w\in U(1),zw\text{is}\text{a}\text{constant}\text{in}U(1)\right\}$E10

is identical to some

$P^{\prime }:=\text{span}((z_1\mathrm{,}{z}_1{e}^{i\theta }),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}{e}^{i\theta }))$E11

where $z_1\mathrm{,}{z}_2\in \mathbb{C}\backslash \left\{0\right\}$. For any $\lambda (e^{i\phi }\mathrm{,}{e}^{i\psi })\in P$, let $z_1=\lambda e^{i\phi }$, $\theta =\psi -\phi$, we have $P=\text{span}((z_1\mathrm{,}{z}_1{e}^{i\theta }),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}{e}^{i\theta }))=P^{\prime }$ where $z_2\in \mathbb{C}\backslash \left\{0\right\}$.

We can get ${\kappa }_1(z_1\mathrm{,0}),(\mathrm{0,}{z}_2))=0$. On the other hand, for any

$(z\mathrm{,}w)={\lambda }_1(z_1\mathrm{,}{z}_1)+{\lambda }_2(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2})\in \text{span}((z_1\mathrm{,}{z}_1),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2})),$E12

where ${\lambda }_1\mathrm{,}{\lambda }_2\in \mathbb{R}$,

$\begin{array}{lll}|w{|}^2\hfill & =\hfill & ({\lambda }_1{z}_1+{\lambda }_2\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2})({\lambda }_1{\displaystyle \overline{z_1}}+{\lambda }_2\frac{{{\displaystyle \overline{z_1}}}^2{z}_2}{|z_1{|}^2})\hfill \\ \hfill & =\hfill & {\lambda }_1^2{z}_1{\displaystyle \overline{z_1}}+{\lambda }_1{\lambda }_2{\displaystyle \overline{z_1}}{z}_2+{\lambda }_2{\lambda }_1{z}_1{\displaystyle \overline{z_2}}+{\lambda }_2^2{\displaystyle \overline{z_2}}{z}_2\hfill \\ \hfill & =\hfill & ({\lambda }_1{z}_1+{\lambda }_2{z}_2)({\lambda }_1{\displaystyle \overline{z_1}}+{\lambda }_2{\displaystyle \overline{z_2}})\hfill \\ \hfill & =\hfill & |z{|}^2\mathrm{,}\hfill \end{array}$E13

that implies $\left|\frac{w}{z}\right|=1$. Therefore, we have

$\begin{array}{lll}{\kappa }_{(z\mathrm{,}w)}((z_1\mathrm{,}{z}_1),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}))\hfill & =\hfill & \frac{3}{2}(Im(z_2{\displaystyle \overline{z_1}})+\frac{3}{2}Im(\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}{\displaystyle \overline{z_1}}))\hfill \\ \hfill & \hfill & -\frac{1}{2}Im(\frac{z}{w}|\frac{w}{z}|(\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}{\displaystyle \overline{z_1}}-z_1{\displaystyle \overline{z_2}}))\hfill \\ \hfill & =\hfill & \frac{3}{2}(Im(z_2{\displaystyle \overline{z_1}})+Im(z_1{\displaystyle \overline{z_2}}))\hfill \\ \hfill & =\hfill & 0.\hfill \end{array}$E14

So $\kappa$ vanishes on $\text{span}((z_1\mathrm{,}{z}_1),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}))$ for any $z_1\mathrm{,}{z}_2\in \mathbb{C}\backslash \left\{0\right\}$, $Im(\frac{z_1}{z_1})\ne 0$.

Conversely, suppose that $\kappa$ vanishes on a plane $P$ spanned by $(z_1\mathrm{,}{w}_1)$ and $(z_2\mathrm{,}{w}_2)$. We know that

$(1+\frac{1}{2}|\frac{w}{z}|)Im(z_2{\displaystyle \overline{z_1}})+(1+\frac{1}{2}|\frac{z}{w}|)Im(w_2{\displaystyle \overline{w_1}})+\frac{1}{2}Im(\frac{z}{w}|\frac{w}{z}|(w_2{\displaystyle \overline{z_1}}-w_1{\displaystyle \overline{z_2}}))=0$E15

holds for any $(z\mathrm{,}w)\in \text{span}((z_1\mathrm{,}{w}_1),(z_2\mathrm{,}{w}_2))$. In the following argument, we divide it into three cases to discuss in terms of $\left|\frac{w}{z}\right|$ and $\frac{w}{z}\left|\frac{w}{z}\right|$.

The first case is that $\left|\frac{w}{z}\right|=\lambda$ for some fixed $\lambda >0$. Let $(z\mathrm{,}w)={\lambda }_1(z_1\mathrm{,}{w}_1)+{\lambda }_2(z_2\mathrm{,}{w}_2)$ for any ${\lambda }_1\mathrm{,}{\lambda }_2\in \mathbb{R}$, then $|{\lambda }_1{w}_1+{\lambda }_2{w}_2|=\lambda |{\lambda }_1{z}_1+{\lambda }_2{z}_2|$, that implies $\left|w_1\right|=\lambda \left|z_1\right|$, $\left|w_2\right|=\lambda \left|z_2\right|$ and $Re(w_1{\displaystyle \overline{w_2}})={\lambda }^{2}Re(z_1{\displaystyle \overline{z_2}})$. It follows that $w_1=\lambda e^{i\theta }{z}_1$, $w_2=\lambda e^{i\theta }{z}_2$, or $w_1=\lambda e^{i\theta }{z}_1$, $w_2=\lambda e^{i\theta }\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}$ for some $\theta \in [\mathrm{0,2}\pi )$.

In the sub-case of $w_1=\lambda e^{i\theta }{z}_1$, $w_2=\lambda e^{i\theta }{z}_2$ for some $\theta \in [\mathrm{0,2}\pi )$, by Eq. (15) we have

$(1+\frac{\lambda }{2})Im(z_2{\displaystyle \overline{z_1}})+(1+\frac{1}{2\lambda }){\lambda }^{2}Im(z_2{\displaystyle \overline{z_1}})+\lambda Im(z_2{\displaystyle \overline{z_1}})=(1+\lambda {)}^{2}Im(z_2{\displaystyle \overline{z_1}})=\mathrm{0,}$E16

which implies $Im(z_2{\displaystyle \overline{z_1}})=0$ and furthermore $Im(w_2{\displaystyle \overline{w_1}})=0$. That means $(z_1\mathrm{,}{w}_1)$ and $(z_2\mathrm{,}{w}_2)$ are colinear. So this case cannot occur.

However, for the other sub-case of $w_1=\lambda e^{i\theta }{z}_1$, $w_2=\lambda e^{i\theta }\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}$ for some $\theta \in [\mathrm{0,2}\pi )$, by Eq. (15) we have

$(1+\frac{\lambda }{2})Im(z_2{\displaystyle \overline{z_1}})+(1+\frac{1}{2\lambda }){\lambda }^{2}Im(z_1{\displaystyle \overline{z_2}})=(1-{\lambda }^2)Im(z_2{\displaystyle \overline{z_1}})=0.$E17

Then $\lambda =1$ or $Im(z_2{\displaystyle \overline{z_1}})=0$, but $(z_1\mathrm{,}{w}_1)$ and $(z_2\mathrm{,}{w}_2)$ cannot be colinear. So, we have $\lambda =1$ which gives

$P=\text{span}((z_1\mathrm{,}{z}_1{e}^{i\theta }),(z_2\mathrm{,}\frac{z_1^2{\displaystyle \overline{z_2}}}{|z_1{|}^2}{e}^{i\theta })),$E18

where $z_1\mathrm{,}{z}_2\in \mathbb{C}\backslash \left\{0\right\}$ and $Im(z_1{\displaystyle \overline{z_2}})\ne 0$ for some $\theta \in [\mathrm{0,2}\pi )$. This finishes the first case.

The second case is $\frac{w}{z}\left|\frac{w}{z}\right|=e^{i\theta }$ for some fixed $\theta \in [\mathrm{0,2}\pi )$. Let $w_1={\lambda }_1{e}^{i\theta }{z}_1\mathrm{,}{w}_2={\lambda }_2{e}^{i\theta }{z}_2$ for some ${\lambda }_1\mathrm{,}{\lambda }_2>0$. Then it follows from (15) that

$\begin{array}{lll}\hfill & \hfill & (1+\frac{{\lambda }_1}{2})Im(z_2{\displaystyle \overline{z_1}})+(1+\frac{1}{2{\lambda }_1}){\lambda }_1{\lambda }_{2}Im(z_2{\displaystyle \overline{z_1}})+\frac{1}{2}({\lambda }_1+{\lambda }_2)Im(z_2{\displaystyle \overline{z_1}})\hfill \\ \hfill & =\hfill & (1+\frac{{\lambda }_2}{2})Im(z_2{\displaystyle \overline{z_1}})+(1+\frac{1}{2{\lambda }_2}){\lambda }_1{\lambda }_{2}Im(z_2{\displaystyle \overline{z_1}})+\frac{1}{2}({\lambda }_1+{\lambda }_2)Im(z_2{\displaystyle \overline{z_1}})\hfill \\ \hfill & =\hfill & (1+{\lambda }_1)(1+{\lambda }_2)Im(z_2{\displaystyle \overline{z_1}})\hfill \\ \hfill & =\hfill & 0\hfill \end{array}$E19

at the points $(z_1\mathrm{,}{w}_1)$ and $(z_2\mathrm{,}{w}_2)$, which implies $Im(z_2{\displaystyle \overline{z_1}})=0$ and furthermore $Im(w_2{\displaystyle \overline{w_1}})=0$. Thus, $z_1$ and $z_2$, $w_1$, and $w_2$ are colinear, which implies that P equals a plane spanned by one vector from $\left\{(z_1\mathrm{,0}),(z_2\mathrm{,0})\right\}$ and the other from $\left\{(\mathrm{0,}{w}_1),(\mathrm{0,}{w}_2)\right\}$. Thus $P\in {\mathbb{T}}^2$.

The last case is the negative to the first one and the second one. It gives $Im(z_2{\displaystyle \overline{z_1}})=Im(w_2{\displaystyle \overline{w_1}})=0$ and $w_2{\displaystyle \overline{z_1}}-w_1{\displaystyle \overline{z_2}}=0$ because of the linear independence, but the former implies the latter by linear transformation, so it is brought down to $Im(z_2{\displaystyle \overline{z_1}})=Im(w_2{\displaystyle \overline{w_1}})=0$. Thus, we have $P\in {\mathbb{T}}^2$ 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 $({\mathbb{R}}^2\mathrm{,}F)$ first, where F is a Finsler metric. The natural symplectic form on $T^{*}{\mathbb{R}}^2$ is $\text{d}x\wedge \text{d}{\displaystyle \overline{\xi }}+\text{d}y\wedge \text{d}{\displaystyle \overline{\eta }}$, and then the natural symplectic form on $T{\mathbb{R}}^2$ induce by the Finsler metric F is

$\begin{array}{c}\omega :=\text{d}x\wedge \text{d}\frac{\partial F}{\partial \xi }+\text{d}y\wedge \text{d}\frac{\partial F}{\partial \eta }\\ =\frac{{\partial }^{2}F}{\partial {\xi }^2}\text{d}x\wedge \text{d}\xi +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\text{d}x\wedge \text{d}\eta +\text{d}y\wedge \text{d}\xi )+\frac{{\partial }^{2}F}{\partial {\eta }^2}\text{d}y\wedge \text{d}\eta \mathrm{.}\end{array}$E20

Define a projection $\pi :T{\mathbb{R}}^2\to {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$ by

$\pi ((x\mathrm{,}y);(\xi \mathrm{,}\eta ))=((x\mathrm{,}y)-\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y))(\xi \mathrm{,}\eta );(\xi \mathrm{,}\eta )).$E21

Let $S_F$ be the unit circle in the Minkowski plane and $T{S}_F$ be its tangent bundle. It is a fact that $T{S}_F\cong {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$. On the other hand, since $T{S}_F$ is embedded in $T{\mathbb{R}}^2$, it inherits a natural symplectic form ${\omega }_0:=\omega {|}_{T{S}_F}$ from $T{\mathbb{R}}^2$.

Theorem 5.1. ${\pi }^{*}{\omega }_0=\omega {|}_{S^{*}{\mathbb{R}}^2}\mathrm{.}$

Proof. Applying the equality

$\frac{\partial F}{\partial \xi }\text{d}\xi +\frac{\partial F}{\partial \eta }\text{d}\eta =\mathrm{0,}$E22

we obtain

$\begin{array}{c}{\pi }^{*}{\omega }_0=\frac{{\partial }^{2}F}{\partial {\xi }^2}\text{d}(x-\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y))\xi )\wedge \text{d}\xi +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\text{d}(x-\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y))\xi )\wedge \text{d}\eta \\ +\text{d}(y-\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y))\eta )\wedge \text{d}\xi )+\frac{{\partial }^{2}F}{\partial {\eta }^2}\text{d}(y-\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y))\eta )\wedge \text{d}\eta \\ =\frac{{\partial }^{2}F}{\partial {\xi }^2}\text{d}x\wedge \text{d}\xi +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\text{d}x\wedge \text{d}\eta +\text{d}y\wedge \text{d}\xi )+\frac{{\partial }^{2}F}{\partial {\eta }^2}\text{d}y\wedge \text{d}\eta \\ -\text{d}(\text{d}F(\xi \mathrm{,}\eta )((x\mathrm{,}y)))\wedge (\frac{{\partial }^{2}F}{\partial {\xi }^2}\xi \text{d}\xi +\frac{{\partial }^{2}F}{\partial {\eta }^2}\eta \text{d}\eta +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\xi \text{d}\eta +\eta \text{d}\xi )).\end{array}$E23

By the positive homogeneity of F, one can get the useful fact that $F(\xi \mathrm{,}\eta )=\xi \frac{\partial F}{\partial \xi }+\eta \frac{\partial F}{\partial \eta }$. Therefore,

$\xi \frac{\partial F}{\partial \xi }+\eta \frac{\partial F}{\partial \eta }=1.$E24

By differentiating (24), we get

$\frac{{\partial }^{2}F}{\partial {\xi }^2}\xi \text{d}\xi +\frac{{\partial }^{2}F}{\partial {\eta }^2}\eta \text{d}\eta +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\xi \text{d}\eta +\eta \text{d}\xi )+\frac{\partial F}{\partial \xi }\text{d}\xi +\frac{\partial F}{\partial \eta }\text{d}\eta =0.$E25

Applying (22) again, we have

$\frac{{\partial }^{2}F}{\partial {\xi }^2}\xi \text{d}\xi +\frac{{\partial }^{2}F}{\partial {\eta }^2}\eta \text{d}\eta +\frac{{\partial }^{2}F}{\partial \xi \partial \eta }(\xi \text{d}\eta +\eta \text{d}\xi )=0.$E26

Thus, the claim follows.

Remark 5.2. For a n-dimensional Minkowski space $({\mathbb{R}}^n\mathrm{,}F)$, we just need to add more indices, then the theorem above is also true for $({\mathbb{R}}^n\mathrm{,}F)$.

Therefore, letting F be a Finsler metric on ${\mathbb{R}}^n$ and S_F be the unit sphere in the Minkowski space $({\mathbb{R}}^n\mathrm{,}F)$, we obtain the following general theorem:

Theorem 5.3. The symplectic form on the space of lines in a Minkowski space $({\mathbb{R}}^n\mathrm{,}F)$ is the canonical symplectic form on the tangent bundle $T{S}_F$ as imbedded in $T{\mathbb{R}}^n$.

We have the following remarks:

Remark 5.4. Theorem 5.3 provides a perspective that we can transform calculus on ${\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$ to ones on $T{S}_F$.

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

$\text{Length}(\gamma )=\frac{1}{4}{\displaystyle {\int }_0^{\infty }}{\displaystyle {\int }_0^{2\pi }}\#(\gamma \cap l(r\mathrm{,}\theta ))\text{d}\theta \text{d}r\mathrm{,}$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 ${\mathbb{R}}^2$ by ${\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$.

As for Minkowski plane, it is a normed two dimensional space with a norm $F(\cdot )=\left|\right|\cdot \left|\right|$, 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

$F(\cdot )={\displaystyle {\int }_{S^1}}|⟨\xi \mathrm{,}\cdot ⟩|g(\theta )\text{d}\theta \mathrm{,}$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, ${\mathbb{R}}^2{\displaystyle \stackrel{{\pi }_1}{\leftarrow }}\mathcal{I}{\displaystyle \stackrel{{\pi }_2}{\to }}{\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$, where $\mathcal{I}:=\left\{(x\mathrm{,}l)\in {\mathbb{R}}^2\times {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}:x\in l\right\}$ is the incidence relations and ${\pi }_1$ and ${\pi }_2$ are projections. A formula one can take as an example of the fundamental theorem of Gelfand transform is the following:

${\displaystyle {\int }_{\gamma }}{\pi }_{1*}{\pi }_2^{*}\left|\Omega \right|={\displaystyle {\int }_{l\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)\left|\Omega \right|,$E29

where $\Omega :=g(\theta )\text{d}\theta \wedge \text{d}r$. However, here we provide a direct proof for this fundamental theorem of Gelfand transform.

Proof. First, consider the case of $\Omega =\text{d}\theta \wedge \text{d}r$. For any $v\in T_x\gamma$, since there is some $v^{\prime }\in T_{x^{\prime }}\mathcal{I}$, such that ${\pi }_{1*}(v^{\prime })=v$, then

$\begin{array}{lll}{({\pi }_{1*}{\pi }_2^{*}|\Omega |)}_x(v)\hfill & =\hfill & {({\displaystyle {\int }_{{\pi }_1^{-1}(x)}}{\pi }_2^{*}|\Omega |)}_x(v)\hfill \\ \hfill & =\hfill & {\displaystyle {\int }_{x^{\prime }\in {\pi }_1^{-1}(x)}}{({\pi }_2^{*}|\Omega |)}_{x^{\prime }}(v^{\prime })\hfill \\ \hfill & =\hfill & {\displaystyle {\int }_{S^1}}({\pi }_2^{*}|\text{d}\theta \wedge \text{d}r|)(v^{\prime })\hfill \\ \hfill & =\hfill & {\displaystyle {\int }_{S^1}}|\text{d}r({\pi }_{2*}(v^{\prime }))|\text{d}\theta \hfill \\ \hfill & =\hfill & {\displaystyle {\int }_{S^1}}|⟨v\mathrm{,}\theta ⟩|\text{d}\theta \hfill \\ \hfill & =\hfill & 4\left|v\right|.\hfill \end{array}$E30

Thus, we have

${\displaystyle {\int }_{\gamma }}{\pi }_{1*}{\pi }_2^{*}\left|\Omega \right|=4\text{Length}(\gamma )={\displaystyle {\int }_{l\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)\left|\Omega \right|$E31

by using the classic Crofton formula.

For the general case of $\Omega =f(\theta )\text{d}\theta \wedge \text{d}r$, we just need to substitute dθ by $g(\theta )\text{d}\theta$ in the equalities in the first case.

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

${\displaystyle {\int }_{\gamma }}{\pi }_{1*}{\pi }_2^{*}\left|\Omega \right|={\displaystyle {\int }_a^b}({\pi }_{1*}{\pi }_2^{*}|\Omega |)({\gamma }^{\prime }(t))\text{d}t={\displaystyle {\int }_a^b}4F({\gamma }^{\prime }(t))\text{d}t=4\text{Length}(\gamma ),$E32

for any curve $\gamma (t):[a\mathrm{,}b]\to {\mathbb{R}}^2$ differentiable almost everywhere in the Minkowski space. Therefore, by using (29), we obtain that

$\text{Length}(\gamma )=\frac{1}{4}{\displaystyle {\int }_{l\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)|g(\theta )\text{d}\theta \wedge \text{d}r|$E33

for Minkowski plane.

The Holmes-Thompson area ${\text{HT}}^2(U)$ of a measurable set U in a Minkowski plane is defined as ${\text{HT}}^2(U):=\frac{1}{\pi }{\displaystyle {\int }_{D^{*}U}}|{\omega }_0{|}^2$, where ω₀ is the natural symplectic form on the cotangent bundle of ${\mathbb{R}}^2$ and $D^{*}U:=\left\{(x\mathrm{,}\xi )\in T^{*}{\mathbb{R}}^2:F^{*}(\xi )\le 1\right\}$. To study it from the perspective of integral geometry, we need to introduce a symplectic form ω to the space of affine lines ${\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$ 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

$\text{Length}(\gamma )=\frac{1}{4}{\displaystyle {\int }_{{\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)\left|\omega \right|.$E34

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

$L:[\mathrm{0,}||p_2-p_2||]\to {\mathbb{R}}^2\mathrm{,}L(t)=p_1+\frac{p_2-p_1}{\left|\right|p_2-p_1\left|\right|}t\mathrm{,}$E35

starting at $p_1$ and ending at $p_2$ in ${\mathbb{R}}^2$. First, using the diffeomorphism between the circle bundle and co-circle bundle, which is

$\begin{array}{c}{\phi }_F:S{\mathbb{R}}^2\to S^{*}{\mathbb{R}}^2\\ {\phi }_F(x\mathrm{,}\xi )=(x\mathrm{,}\text{d}{F}_{\xi }),\end{array}$E36

we can obtain a fact that

$\begin{array}{l}{\displaystyle {\int }_{L\times \left\{\frac{p_2-p_1}{\Vert p_2-p_1\Vert }\right\}}{\phi }_F^{*}{\alpha }_0}={\displaystyle {\int }_{{\phi }_F(L\times \left\{\frac{p_2-p_1}{\Vert p_2-p_1\Vert }\right\})}{\alpha }_0}\hfill \\ ={\displaystyle {\int }_0^{\left|\right|p_2-p_1\left|\right|}}{\alpha }_{0\text{d}{F}_{\frac{p_2-p_1}{\left|\right|p_2-p_1\left|\right|}}}((\frac{p_2-p_1}{\left|\right|p_2-p_1\left|\right|},0))\text{d}t\hfill \\ ={\displaystyle {\int }_0^{\left|\right|p_2-p_1\left|\right|}}\text{d}{F}_{\frac{p_2-p_1}{\left|\right|p_2-p_1\left|\right|}}(\frac{p_2-p_1}{\left|\right|p_2-p_1\left|\right|})\text{d}t\mathrm{,}\hfill \end{array}$E37

where ${\alpha }_0$ is the tautological one-form, precisely ${\alpha }_{0\xi }(X):=\xi ({\pi }_{0*}X)$ for any $X\in T_{\xi }{T}^{*}{\mathbb{R}}^2$, and $\text{d}{\alpha }_0={\omega }_0$. Applying the basic equality that $\text{d}{F}_{\xi }(\xi )=1$, which is derived from the positive homogeneity of F, for all $\xi \in S{\mathbb{R}}^2$, the above quantity becomes ${\displaystyle {\int }_0^{\left|\right|p_2-p_1\left|\right|}}1\text{d}t$, which equals $\left|\right|p_2-p_1\left|\right|$.

Let $R:=\left\{{\xi }_x\in S^{*}{\mathbb{R}}^2:x\in {\displaystyle \overline{p_1{p}_2}}\right\}$ and $T=\left\{l\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}:l\cap {\displaystyle \overline{p_1{p}_2}}\ne Ø\right\}$, and p′ is the projection (composition) from $S^{*}{\mathbb{R}}^2$ to ${\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}$.

Apply the above fact and ${p^{\prime }}^{*}\omega ={\omega }_0$,

$\begin{array}{lllllll}{\displaystyle {\int }_T}\left|\omega \right|\hfill & =\hfill & {\displaystyle {\int }_{p^{\prime }(R)}}\left|\omega \right|\hfill & =\hfill & {\displaystyle {\int }_R}\left|{p^{\prime }}^{*}\omega \right|\hfill & =\hfill & {\displaystyle {\int }_R}\left|{\omega }_0\right|\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & =\hfill & \left|{\displaystyle {\int }_{R^{+}}}{\omega }_0\right|+\left|{\displaystyle {\int }_{R^{-}}}{\omega }_0\right|\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & =\hfill & \left|{\displaystyle {\int }_{\partial R^{+}}}{\alpha }_0\right|+\left|{\displaystyle {\int }_{\partial R^{-}}}{\alpha }_0\right|\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & =\hfill & 4\left|\right|p_2-p_1\left|\right|.\hfill \end{array}$E38

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

Furthermore, combining with (33), we have

$\frac{1}{4}{\displaystyle {\int }_{l\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)\left|\Omega \right|=\frac{1}{4}{\displaystyle {\int }_{{\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}}}\#(\gamma \cap l)\left|\omega \right|,$E39

where $\Omega =g(\theta )\text{d}\theta \wedge \text{d}r$. Then, by the injectivity of cosine transform in Ref. [24], $\left|\Omega \right|=\left|\omega \right|$.

To obtain the HT area, one can define a map

$\begin{array}{c}\pi :{\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\times {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\backslash \Delta \to {\mathbb{R}}^2\\ \pi (l\mathrm{,}{l}^{\prime })=l\cap l^{\prime }\mathrm{,}\end{array}$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

${\text{HT}}^2(U)=\frac{1}{2\pi }{\displaystyle {\int }_{x\in {\mathbb{R}}^2}}\chi (x\cap U)\left|{\pi }_{*}{\Omega }^2\right|\mathrm{.}$E41

Proof. On the one hand,

$\frac{1}{\pi }{\displaystyle {\int }_{D^{*}U}}{\omega }_0^2=\frac{1}{\pi }{\displaystyle {\int }_{\partial D^{*}U}}{\omega }_0^2=\frac{1}{\pi }{\displaystyle {\int }_{S^{*}U}}{\alpha }_0\wedge {\omega }_0\mathrm{.}$E42

On the other hand,

$\begin{array}{lll}\frac{1}{\pi }{\displaystyle {\int }_{x\in {\mathbb{R}}^2}}\chi (x\cap U){\pi }_{*}{\Omega }^2\hfill & =\hfill & \frac{1}{\pi }{\displaystyle {\int }_{\left\{(l\mathrm{,}{l}^{\prime })\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\times {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\backslash \Delta :l\cap l^{\prime }\in U\right\}}}{\Omega }^2\hfill \\ \hfill & =\hfill & \frac{1}{\pi }{\displaystyle {\int }_{\left\{(l\mathrm{,}{l}^{\prime })\in {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\times {\displaystyle \overline{G{r}_1({\mathbb{R}}^2)}}\backslash \Delta :l\cap l^{\prime }\in U\right\}}}{\omega }^2\hfill \\ \hfill & =\hfill & \frac{1}{\pi }{\displaystyle {\int }_{{\mathbb{T}}^{*}U\backslash \left\{(x\mathrm{,}\xi \mathrm{,}\xi ):\xi \in S_x^{*}U\right\}}}{p^{\prime }}^{*}{\omega }^2\hfill \\ \hfill & =\hfill & \frac{1}{\pi }{\displaystyle {\int }_{{\mathbb{T}}^{*}U\backslash \left\{(x\mathrm{,}\xi \mathrm{,}\xi ):\xi \in S_x^{*}U\right\}}}{\omega }_0^2\hfill \\ \hfill & =\hfill & \frac{2}{\pi }{\displaystyle {\int }_{\left\{(x\mathrm{,}\xi \mathrm{,}\xi ):\xi \in S_x^{*}U\right\}}}{\alpha }_0\wedge {\omega }_0\hfill \\ \hfill & =\hfill & \frac{2}{\pi }{\displaystyle {\int }_{S^{*}U}}{\alpha }_0\wedge {\omega }_0\mathrm{,}\hfill \end{array}$E43

where

${\mathbb{T}}^{*}U:=\left\{(x\mathrm{,}\xi \mathrm{,}{\xi }^{\prime }):\xi \mathrm{,}{\xi }^{\prime }\in S_x^{*}U\right\}\mathrm{.}$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.