皮特·奥立弗著的《等值不变量和对称性（英文版）》主要介绍了等价、不变量和对称等相关知识，具体为几何基础、李群变换、表示理论、射流与接触变换、微分不变量、微分方程的对称性、变分问题的对称性、等价问题的提法、嘉当等价法、微分系统、Frobenius定理、Cartan-Kahler存在性定理等内容，内容丰富，条理清晰。
本书内容涉及多个数学学科，包括几何、分析、应用数学和代数，提出了一种创新的方法，用于研究在各种数学领域和物理应用中出现的等价和对称问题。建立了求解等价问题的建设性方法，并应用于各种数学学科，包括微分方程、变分问题、流形、黎曼矩阵、多项式和微分算子。

Preface
Acknowledgments
Introduction
1. Geometric Foundations
Manifolds
Functions
Submanifolds
Vector Fields
Lie Brackets
The Differential
Differential Forms
Equivalence of Differential Forms
2. Lie Groups
Transformation Groups
Invariant Subsets and Equations
Canonical Forms
Invariant Functions
Lie Algebras
Structure Constants
The Exponential Map
Subgroups and Subalgebras
Infinitesimal Group Actions
Classification of Group Actions
Infinitesimal Invariance
Invariant Vector Fields
Lie Derivatives and Invariant Differential Forms
The Maurer-Cartan Forms
3. Representation Theory
Representations
Representations on Function Spaces
Multiplier Representations
Infinitesimal Multipliers
Relative Invariants
Classical Invariant Theory
4. Jets and Contact Transformations
Transformations and Functions
Invariant Functions
Jets and Prolongations
Total Derivatives
Prolongation of Vector Fields
Contact Forms
Contact Transformations
Infinitesimal Contact Transformations
Classification of Groups of Contact Transformations
5. Differential Invariants
Differential Invariants
Dimensional Considerations
Infinitesimal Methods
Stabilization and Effectiveness
Invariant Differential Operators
Invariant Differential Forms
Several Dependent Variables
Several Independent Variables
6. Symmetries of Differential Equations
Symmetry Groups and Differential Equations
Infinitesimal Methods
Integration of Ordinary Differential Equations
Characterization of Invariant Differential Equations
Lie Determinants
Symmetry Classification of Ordinary Differential Equations
A Proof of Finite Dimensionality
Linearization of Partial Differential Equations
Differential Operators
Applications to the Geometry of Curves
7. Symmetries of Variational Problems
The Calculus of Variations
Equivalence of Functionals
Invariance of the Euler-Lagrange Equations
Symmetries of Variational Problems
Invariant Variational Problems
Symmetry Classification of Variational Problems
First Integrals
The Cartan Form
Invariant Contact Forms and Evolution Equations
8. Equivalence of Coframes
Frames and Coframes
The Structure Functions
Derived Invariants
Classifying Functions
The Classifying Manifolds
Symmetries of a Coframe
Remarks and Extensions
9. Formulation of Equivalence Problems
Equivalence Problems Using Differential Forms
Coframes and Structure Groups
Normalization
Overdetermined Equivalence Problems
10. Cartan's Equivalence Method
The Structure Equations
Absorption and Normalization
Equivalence Problems for Differential Operators
Fiber-preserving Equivalence of Scalar Lagrangians
An Inductive Approach to Equivalence Problems
Lagrangian Equivalence under Point Transformations
Applications to Classical Invariant Theory
Second Order Variational Problems
Multi-dimensional Lagrangians
11. Involution
Cartan's Test
The Transitive Case
Divergence Equivalence of First Order Lagrangians
The Intrinsic Method
Contact Transformations
Darboux' Theorem
The Intransitive Case
Equivalence of Nonclosed Two-Forms
12. Prolongation of Equivalence Problems
The Determinate Case
Equivalence of Surfaces
Conformal Equivalence of Surfaces
Equivalence of Riemannian Manifolds
The Indeterminate Case
Second Order Ordinary Differential Equations
13. Differential Systems
Differential Systems and Ideals
Equivalence of Differential Systems
Vector Field Systems
14. Frobenius' Theorem
Vector Field Systems
Differential Systems
Characteristics and Normal Forms
The Technique of the Graph
Global Equivalence
15. The Cartan-Kaihler Existence Theorem
The Cauchy-Kovalevskaya Existence Theorem
Necessary Conditions
Sufficient Conditions
Applications to Equivalence Problems
Involutivity and Transver