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 R<sup>2</sup> with Lp norm, 1 ≤ p < ∞, 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.

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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<sup>p</sup> space, 1 < p < ∞, has the main features of a Finsler space. As such, we focus on L<sup>p</sup> 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 L<sup>p</sup> 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 < p ≤ 1, 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. [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<sup>p</sup> 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:

$$\mathbb{R}^2 \xleftarrow{n\_1} \mathbb{F} \xrightarrow{n\_2} \overline{Gr\_1(\mathbb{R}^2)}\tag{1}$$

where

$$\begin{aligned} \mathbf{F} &= \left\{ ((\mathbf{x}, \mathbf{y}), l(r, \theta)) : (\mathbf{x}, \mathbf{y}) \in \mathbf{R}^2, l(r, \theta) \in \overline{Gr\_1(\mathbf{R}^2)}, (\mathbf{x}, \mathbf{y}) \in l(r, \theta) \right\}, \\ &\simeq \left\{ (\mathbf{x}, \mathbf{y}, r, \theta) : \mathbf{x} \cos \left(\theta\right) + \mathbf{y} \sin \left(\theta\right) = r \right\}, \end{aligned}$$

π<sup>1</sup> and π<sup>2</sup> are the natural projections of fibers. The Gelfand transform of a 2-density ϕ ¼ jdr∧dθj is defined as

$$\text{GT}(\varphi) = \pi\_{1\_\*} \pi\_2^\* \varphi,\tag{2}$$

which is a 1-density R<sup>2</sup> .
