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Differential Geometry

Year 3 · 5 sections · 61 concepts · 133 practice questions

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What's covered

free1 · Curves in the plane and in space13 concepts

Overview 1 · The plan of this section
Definition 1.1 · Regular curves and arc length
Proposition 1.2 · Curvature in any parametrisation
Theorem 1.3 · The Frenet–Serret equations
Proposition 1.4 · Torsion in any parametrisation
Corollary 1.5 · Zero torsion means planar
Theorem 1.6 · The fundamental theorem of curves
Definition 1.7 · Signed curvature of plane curves
Theorem 1.8 · The rotation index (Hopf's Umlaufsatz)
Lemma 1.9 · Wirtinger's inequality
Theorem 1.10 · The isoperimetric inequality
Theorem 1.11 · The four vertex theorem
Check 1.12 · Check: computing curvature and torsion

2 · Surfaces and the first fundamental form12 concepts

Overview 2 · The plan of this section
Definition 2.1 · Regular surfaces and charts
Proposition 2.2 · Level sets are surfaces
Proposition 2.3 · The tangent plane
Example 2.4 · Standard surfaces
Definition 2.5 · The first fundamental form
Proposition 2.6 · Lengths, angles and areas
Proposition 2.7 · Change of coordinates
Definition 2.8 · Isometries and local isometries
Proposition 2.9 · Isometry in coordinates
Definition 2.10 · Conformal maps
Check 2.11 · Check: working with the first fundamental form

3 · Curvature of surfaces12 concepts

Overview 3 · The plan of this section
Definition 3.1 · The Gauss map and the Weingarten map
Proposition 3.2 · The second fundamental form
Proposition 3.3 · Normal curvature and Meusnier's theorem
Theorem 3.4 · Principal curvatures and Euler's formula
Proposition 3.5 · Gaussian and mean curvature
Definition 3.6 · Elliptic, hyperbolic, parabolic and umbilic points
Proposition 3.7 · Curvature of a graph
Proposition 3.8 · Curvature of a surface of revolution
Proposition 3.9 · Totally umbilic surfaces
Theorem 3.10 · Compact surfaces have an elliptic point
Check 3.11 · Check: computing K and H

4 · The Theorema Egregium11 concepts

Overview 4 · The plan of this section
Definition 4.1 · Christoffel symbols
Proposition 4.2 · Christoffel symbols depend only on the metric
Theorem 4.3 · The Gauss and Codazzi–Mainardi equations
Theorem 4.4 · Gauss's Theorema Egregium
Proposition 4.5 · Curvature from the metric
Corollary 4.6 · No perfect flat map of the Earth
Example 4.7 · The catenoid and the helicoid
Theorem 4.8 · Bonnet's theorem (fundamental theorem of surfaces)
Example 4.9 · The hyperbolic plane
Check 4.10 · Check: intrinsic or extrinsic?

5 · Geodesics and the Gauss–Bonnet theorem13 concepts

Overview 5 · The plan of this section
Definition 5.1 · Geodesic curvature
Definition 5.2 · Geodesics
Proposition 5.3 · The geodesic equations
Theorem 5.4 · Existence and uniqueness of geodesics
Proposition 5.5 · Clairaut's relation
Example 5.6 · Geodesics on the sphere and the torus
Theorem 5.7 · The local Gauss–Bonnet theorem
Corollary 5.8 · Angle sums of geodesic triangles
Definition 5.9 · The Euler characteristic
Theorem 5.10 · The global Gauss–Bonnet theorem
Corollary 5.11 · Consequences of Gauss–Bonnet
Check 5.12 · Check: applying Gauss–Bonnet

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Proposition 1.2 · Curvature in any parametrisation

$\kappa = |\gamma''|$ (unit speed) $= |\gamma'\times\gamma''|/|\gamma'|^3$ (any speed). $\kappa\equiv0$ iff a straight line.

Proof

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