Differential Geometry
M13 — Spring 24
Contents
1 Fundamental notions
1.1 Points, vectors and the tangent space
1.2 Smooth maps, diffeomorphisms and the differential
1.3 Vector fields and the gradient
1.4 The cotangent space and the exterior derivative
2 Curves
2.1 Definitions and examples
2.2 Unit speed curves
2.3 Curves in the plane
2.4 Local geometric properties of plane curves
2.5 Global geometric properties of plane curves
2.6 Curves in three-dimensional space
3 Surfaces
3.1 Embedded surfaces
3.2 Tangent planes
3.3 Orientation
3.4 Geodesics
3.5 Covariant derivative
3.6 Curvature of embedded surfaces
3.7 Local parametrisations
3.8 Calculations in local parametrisations
3.9 Immersed surfaces
4 Intrinsic surface geometry
4.1 The Gauss–Codazzi equations
4.2 The covariant derivative revisited
4.3 Curvature tensor and the Theorema Egregium
4.4 Geodesic curvature
4.5 First version of the Gauss–Bonnet Theorem
4.6 Second version of the Gauss–Bonnet Theorem
4.7 Global version of the Gauss–Bonnet Theorem
5 Further topics
5.1 Differential forms
5.2 The Theorema Egregium revisited
Differential Geometry
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