Differential Geometry: Curves and Surfaces

How curves and surfaces bend, and what an inhabitant can measure

About this course

How curves and surfaces bend, how to measure distance on them, and how local geometry shapes paths and global structure — Frenet frames, fundamental forms, curvature, geodesics, the Theorema Egregium and Gauss–Bonnet.

What you will be able to do

  • Compute arc length, curvature and torsion, and use the Frenet frame
  • Compute the first fundamental form and use it to measure lengths and areas on a surface
  • Compute Gaussian and mean curvature from the shape operator for familiar surfaces
  • Find geodesics and describe parallel transport with the covariant derivative
  • Apply the Theorema Egregium and Gauss–Bonnet to relate curvature to global shape

What this course covers

  • Surface charts and the tangent plane
  • Tangent plane and path velocities via the chain rule
  • Coordinate relabelling and normal orientation
  • Chart boundaries versus surface smoothness; atlas idea
  • Tangent plane on a saddle surface
  • Unit Normal Field and Its Construction
  • Shape Operator Definition and Self-Adjointness
  • Principal Curvatures and Euler's Formula
  • Sign Convention, Units, and Orientation Dependence
  • Reparameterization and speed dependence
  • Arc length as a geometric parameter
  • Signed curvature of plane curves
  • Intrinsic vs. extrinsic geometry: plane and cylinder patch
  • Gauss-Bonnet Theorem: Statement and Terms
  • Geodesic Curvature and Corner Angles
  • Curvature Budget: Disc, Cap, and Hemisphere
  • Angle Excess, Closed Surfaces, and Euler Characteristic
  • Geodesics: definition and variational foundation
  • Energy functional and the geodesic equations
  • Zero tangential acceleration characterization
  • Geodesics on a sphere: great circles and global shortest paths
  • First Fundamental Form: Definition and Construction
  • Positive Definiteness and Metric Determinant
  • Paraboloid Metric: Position-Dependent Coefficients and the Cross Term
  • Cylinder and Sphere Metrics: Intrinsic Flatness and Coordinate Failure
  • Sphere metric and directional scale factors
  • Tissot indicatrix as a distortion gauge
  • Three cylindrical projections compared
  • Theorema Egregium and limits of flat maps
  • Maps, meshes, and the underlying geometry
  • Gaussian and Mean Curvature: Definitions and Operator Invariance
  • Pointwise Sign Classification of Surface Points
  • Curvature of Four Canonical Surfaces
  • Graph Formulas for K and H, and Recovery of Principal Curvatures
  • Regularity of a parametrization: definition and ellipse baseline
  • Reparametrization and the stalled-clock false alarm
  • Genuine cusps: obstruction to a regular two-sided chart
  • Regular crossings and the four-level classification framework
  • Covariant derivative via tangent projection
  • Parallel transport: definition and properties
  • Geodesics as self-parallel paths
  • Holonomy on the sphere and intrinsic curvature
  • Plane and sphere as curvature calibrations
  • Cylinder: extrinsic bending without intrinsic curvature
  • Cone: intrinsic flatness and genuine apex singularity
  • Torus: elliptic, parabolic, and hyperbolic regions on one surface
  • Curvature atlas: comparative summary and sign-convention warnings
  • Arc length parameterization of a plane curve
  • Unit tangent vector and curvature of a plane curve
  • Osculating circle and second-order contact
  • Effect of reversal and integration of curvature
  • Intrinsic geometry and Theorema Egregium
  • Surface-of-revolution metric and connection coefficients
  • Intrinsic curvature formula K = −F″/F
  • Coordinate singularities vs. genuine geometric singularities
  • Local vs. global isometry and map-distortion consequences
  • Helix length, curvature, torsion, and geodesic test
  • Sphere geometry: geodesics, Gauss-Bonnet, and holonomy
  • Torus Gaussian curvature and Gauss-Bonnet cancellation
  • Flat-map distortion and intrinsic curvature trade-offs
  • The Frenet moving frame (T, N, B)
  • Curvature of a space curve
  • Torsion and the Frenet-Serret equations
  • Effect of handedness reversal vs. traversal reversal on torsion
  • Unwrapped tangent angle and turning number
  • Calibration with circle traversals (forward, backward, twice)
  • Ellipse and figure-eight: varying curvature and signed cancellation
  • Turning tangent theorem and its precise hypotheses

Syllabus

  1. Curves

    1. L1 · Geometry from Motion and Measurement
    2. L2 · Parametrized Curves and Regularity
    3. L3 · Arc Length and Curvature
    4. L4 · Torsion and the Frenet Frame
    5. L5 · Plane Curves and Global Turning

  2. Surfaces and the First Fundamental Form

    1. L6 · Surfaces, Charts and Tangent Planes
    2. L7 · The First Fundamental Form

  3. Curvature of Surfaces

    1. L8 · Normals and the Shape Operator
    2. L9 · Gaussian and Mean Curvature
    3. L10 · Curvature of Familiar Surfaces

  4. Intrinsic Geometry

    1. L11 · Geodesics and Variational Length
    2. L12 · Covariant Derivatives and Parallel Transport
    3. L13 · Intrinsic Curvature and Gauss's Theorema Egregium

  5. Global Geometry and Surface Models

    1. L14 · Gauss–Bonnet and Global Shape
    2. L15 · Maps, Distortion and Surface Models
    3. L16 · A Geometric Atlas and Capstone