Differential Geometry

Preliminaries
	Categories and Functors
	Linear Algebra
		Linear Spaces and Linear Maps
		Determinant of Matrices and Linear Endomorphisms
		Orientation of a Linear Space
		Tensor Product
		Exterior Powers
		Euclidean Linear Spaces
		Hodge Star Operator
	Geometry
		Affine Geometry
		Euclidean Spaces
	Topology
		Separation and Countability Axioms
		Compactness
		Fundamental Group and Covering Spaces
	Multivariable Calculus
	Measure and Integration
	Ordinary Differential Equations
Curves in En
	The Notion of a Curve
	The Length of a Curve
	Crofton\'s Formula
	The Osculating k-planes
	Frenet Frames and Curvatures
	Limits of Some Geometrical Quantities
	Osculating Spheres
	Plane Curves
		Evolute, Involute, Parallel Curves
		The Rotation Number Theorem
		Convex Curves
		The Four Vertex Theorem
	Curves in R3
		Orthogonal Projections onto Planes Spanned by Frenet Vectors
		Fenchel\'s Theorem
		The Fáry-Milnor Theorem
Hypersurfaces in Rn
	General Theory
		Definition of a Parameterized Hypersurface
		Curvature of Curves on a Hypersurface
		The Weingarten Map and the Fundamental Forms
		Umbilical Points
		The Fundamental Equations of Hypersurface Theory
		Surface Volume
	Surfaces in R3
		Surfaces of Revolution
		Lines of Curvature, Triply Orthogonal Systems
		Ruled and Developable Surfaces
		Asymptotic Curves on Negatively Curved Surfaces
		Surfaces of Constant Negative Curvature
		Minimal Surfaces
Manifolds
	Topological and Differentiable Manifolds
		Basic Definitions
		Configuration Spaces
		Submanifolds of Rn
		Remarks on the Classification of Manifolds
	The Tangent Bundle
		The Tangent Bundle
		The Derivative of a Smooth Map
	The Lie Algebra of Vector Fields
		The Lie Algebra of Lie Groups
		Subspace Distributions and the Frobenius Theorem
	Tensor Bundles and Tensor Fields
	The Lie Derivative
	Differential Forms
		Interior Product by a Vector Field
		Exterior Differentiation
		De Rham Cohomology
	Integration of Differential Forms
		Integration on Chains
		Integration on Regular Domains
		Integration on Riemannian Manifolds
	Differentiation of Vector Fields
	Curvature
	Decomposition of Algebraic Curvature Tensors
	Conformal Invariance of the Weyl Tensor
	Geodesics
	Applications to Hypersurface Theory
		Geodesic Curves on Hypersurfaces
		Clairaut\'s Theorem
		Moving Orthonormal Frames Along a Hypersurface
		Relation to Earlier Formulae for Parameterized Hypersurfaces
		The Gauss–Bonnet Formula
		Steiner\'s Formula
		Minkowski\'s Formula
		Rigidity of Convex Surfaces
Bibliography
Index