FOLLOW AN IDEA
Differential Geometry
Explore curves and surfaces; metrics and curvature; manifolds and forms. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Coordinate geometry
Coordinates translate geometric conditions into equations and algebraic calculations.
Video explanations to be added
Curves and surfaces
Regular curves
A regular curve has a nonzero velocity, allowing local tangent geometry.
Video explanations to be addedArc length
Integrating speed measures a sufficiently regular curve's length.
Video explanations to be addedCurvature and torsion
Curvature measures tangent turning; torsion describes departure from a plane in space.
Video explanations to be addedRegular surfaces
Regular surface charts have linearly independent coordinate tangent vectors.
Video explanations to be added
Metrics and curvature
Tangent spaces
Tangent vectors at a point form a linear space describing infinitesimal directions.
Video explanations to be addedRiemannian metrics
A smoothly varying positive-definite inner product on tangent spaces measures lengths and angles.
Video explanations to be addedGeodesics
Geodesics have zero covariant acceleration and locally extremize length under suitable conditions.
Video explanations to be addedGaussian curvature
The product of principal curvatures measures intrinsic surface curvature.
Video explanations to be added
Manifolds and forms
Smooth manifolds
Compatible smooth coordinate charts support differentiation on a manifold.
Video explanations to be addedDifferential forms
Alternating multilinear objects encode integration over oriented geometric pieces.
Video explanations to be addedCovariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.
Video explanations to be addedGauss–Bonnet theorem
For a closed oriented surface, total Gaussian curvature relates to its Euler characteristic.
Video explanations to be added