该书从掌握经济领域相关研究所需的共形几何理论基础开始，由基础微分几何逐步展开至黎曼几何，严谨、细致地对这一架构下的Geometry of Manifolds（流形几何）、双曲几何（罗巴切夫斯基几何）、微分拓扑曲面、黎曼曲面等进行了系统阐述。同时，还引入在经济尤其是金融领域的研究问题求索中运用此系列数学方法的应用场景，从微分几何在经济尤其是金融领域的应用，推进至共形几何在经济尤其是金融领域的应用，对SABR模型、无套利几何、随机时间下的共形模型等进行了深入讨论。

朱南丽，女，博士，副教授，硕士生导师，MBA导师，现任教于云南财经大学。研究领域涉及互联网金融、管理信息系统等。主持并参与国家自然科学基金等国家及省部级项目多项。以第一作者身份发表被EI收录、被CSSCI收录、北大核心期刊、中国科技核心期刊论文10余篇。获得省、市、校等多级荣誉与奖项，参编教材多部。曾在中国香港、英国等地进行学术访问与交流。

1 Introduction
2 Basic Differential Geometry
2.1 Fields on open sets in real vector spaces
2.2 Closed forms are locally exact
2.3 Fixed Point Theorems
2.4 The abstract field C versus the R-algebra C of complex numbers
2.5 Coordinates and local smooth Rigidity Theorems
2.6 Differentiation in Banaeh spaces
2.7 Sard's Theorem
2.8 Morse Lemma,Morse functions
3 Geometry of Manifolds
3.1 Differential manifolds
3.2 Fields on manifolds
3.3 Integrability condition of Frobenius
3.4 Foliations on manifolds
3.5 Topology of connected,compact surfaces
4 Hyperbolic Geometry
4.1 Hyperbolic plane H=HI
4.2 Intermezzo:Higher Cross Ratios
4.3 Hyperbolic trigonometry
4.4 Hyperbolic Area
4.5 A compact hyperbolic surface of genus g≥2
4.6 The Riemann Sphere □(数理化公式)
5 Examples of Geometry
5.1 The Space of Norms
5.2 Combinatorial Geometry
5.3 Spaces of Involutions
6 Differential Topology of Surfaces
6.1 o-and 1-deRham cohomology of surfaces
6.2 Hyperbolic plane again, now H=HJ
6.3 Reminder:Multi-Linear Algebra
6.4 Reminder:Holomorphie functions in one complex variable
6.5 J-Laplace operator and metric
6.6 J-surfaces
7 Riemann Surfaces
7.1 Riemann surfaces as z-and as J-surface
7.2 Natural structures on the space (TS)
7.3 J-fields and Integrability in higher dimension
7.4 Integrability of fibred J-fields
7.5 Analysis of Laplace operators on J-surfaces
7.6 Topology of the two point Green function
8 Differential Geometry in Economics and Finance
8.1 SABR model
8.2 Geometry of no arbitrage
9 Conformal Geometry in Economics and Finance
9.1 Models with Zero Instantaneous Spot Rate
9.2 Conformal Symmetries and Lorentz Transformations
9.3 Conformal Diffusion
9.4 Time Homogeneity and Hyperbolic (Lorentz) Evolution
9.5 The Conformal Model under Stochastic Clock
References
后记 (Postscript)