目录

●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 )