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包含了陳省身先生有關微分幾何文章的選集以及他在普林斯頓高等研究院的一些講義，大部分未公開出版或是只在小范圍內發表過。陳省身是現代微分幾何之父，本書給讀者展示了微分幾何與其他學科如拓撲學和李群聯系的廣闊前景，作者對各個學科聯系的把握非常精准並且正中要點。
陳省身曾在《Atiyah選集》的前言中說過：「無論新的東西如何被改進或者精化，但最原始的文章總是最直接和最達要點……」本書對想學習現代微分幾何的初學者非常有價值，也對專家們重新思考微分幾何有益。
1 From Triangles to Manifolds
1.1 Geometry
1.2 Triangles
1.3 Curves in the plane; rotation index and regular homotopy
1.4 Euclidean three-space
1.5 From coordinate spaces to manifolds
1.6 Manifolds; local tools
1.7 Homology
1.8 Vector fields and generalizations
1.9 Elliptic differential equations
1.10 Euler characteristic as a source of global invariants
1.11 Gauge field theory
1.12 Concluding remarks
2 Topics in Differential Geometry
2.1 General notions on differentiable manifolds
2.1.1 Homology and cohomology groups of an abstract complex
2.1.2 Product theory
2.1.3 An example
2.1.4 Algebra of a vector space
2.1.5 Differentiable manifolds
2.1.6 Multiple integrals
2.2 Riemannian manifolds
2.2.1 Riemannian manifolds in Euclidean space
2.2.2 Imbedding and rigidity problems in Euclidean space
2.2.3 Affine connection and absolute differentiation
2.2.4 Riemannian metric
2.2.5 The Gauss-Bonnet formula
2.3 Theory of connections
2.3.1 Resume on fiber bundles
2.3.2 Connections
2.3.3 Local theory of connections; the curvature tensor
2.3.4 The homomorphism h and its independence of connection
2.3.5 The homomorphism h for the universal bundle
2.3.6 The fundamental theorem
2.4 Bundles with the classical groups as structural groups
2.4.1 Homology groups of Grassmann manifolds
2.4.2 Differential forms in Grassmann manifolds
2.4.3 Multiplicative properties of the cohomology ring of a Grassmann manifold
2.4.4 Some applications
2.4.5 Duality theorems
2.4.6 An application to projective differential geometry
3 Curves and Surfaces in Euclidean Space
3.1 Theorem of turning tangents
3.2 The four-vertex theorem
3.3 Isoperimetric inequality for plane curves
3.4 Total curvature of a space curve
3.5 Deformation of a space curve
3.6 The Gauss-Bonnet formula
3.7 Uniqueness theorems of Cohn-Vossen and Minkowski
3.8 Bernstein’’s theorem on minimal surfaces
4 Minimal Submanifolds in a Riemannian Manifold
4.1 Review of Riemannian geometry
4.2 The first variation
4.3 Minimal submanifolds in Euclidean space
4.4 Minimal surfaces in Euclidean space
4.5 Minimal submanifolds on the sphere
4.6 Laplacian of the second fundamental form
4.7 Inequality of Simons
4.8 The second variation
4.9 Minimal cones in Euclidean space
5 Characteristic Classes and Characteristic Forms
5.1 Stiefel-Whitney and Pontrjagin classes
5.2 Characteristic classes in terms of curvature
5.3 Transgression
5.4 Holomorphic line bundles and the Nevanlinna theory
6 Geometry and Physics
6.1 Euclid
6.2 Geometry and physics
6.3 Groups of transformations
6.4 Riemannian geometry
6.5 Relativity
6.6 Unified field theory
6.7 Weyl’’s abelian gauge field theory
6.8 Vector bundles
6.9 Why Gauge theory
7 The Geometry of G-Structures
7.1 Introduction
7.2 Riemannian structure
7.3 Connections
7.4 G-structure
7.5 Harmonic forms
7.6 Leaved structure
7.7 Complex structure
7.8 Sheaves
7.9 Characteristic classes
7.10 Riemann-Roch, Hirzebruch, Grothendieck, and Atiyah-Singer Theorems
7.11 Holomorphic mappings of complex analytic manifolds i
7.12 Isometric mappings of Riemannian manifolds
7.13 General theory of G-structures
