Curves, surfaces, manifolds, and the geometry of smooth spaces
← Back to MathematicsStudy parametric curves, curvature, and fundamental properties in ℝ² and ℝ³.
Learn the local theory of surfaces embedded in three-dimensional space.
Study geometric properties that depend only on the surface itself.
Learn the foundations of smooth manifold theory and coordinate systems.
Study vector fields, differential forms, and exterior calculus on manifolds.
Study manifolds equipped with Riemannian metrics and their geometric properties.
Explore the deep connections between curvature and topological properties.
Study continuous symmetry groups and their infinitesimal structure.
Learn about fiber bundles and connections in modern differential geometry.
Explore modern developments and applications of differential geometry.
Study the differential geometry of curves in ℝ² and ℝ³, building intuition for higher-dimensional concepts.
Learn curves as functions γ: I → ℝⁿ. Study regular curves where γ'(t) ≠ 0, and compute arc length s = ∫|γ'(t)|dt.
Define tangent vector T(t) = γ'(t) and unit tangent vector T(t) = γ'(t)/|γ'(t)|. Understand geometric interpretation as velocity and direction.
Learn curvature κ(t) = |T'(t)|/|γ'(t)| measuring how fast the curve turns. Study radius of curvature R = 1/κ and osculating circle.
Derive T' = κN, N' = -κT + τB, B' = -τN for the moving frame {T, N, B} along space curves. Understand geometric significance.
Define principal normal N = T'/|T'| and binormal B = T × N. Learn how these form an orthonormal frame called the Frenet frame.
Study torsion τ = (γ' × γ'' · γ''')/ |γ' × γ''|² measuring how much the curve twists out of its osculating plane.
Learn that curvature κ(s) and torsion τ(s) uniquely determine a curve up to rigid motions. Understand reconstruction from invariants.
Study helix (constant curvature and torsion), cycloid (generated by rolling circle), catenary (hanging chain), and their geometric properties.
Learn the local differential geometry of surfaces embedded in three-dimensional Euclidean space.
Study surfaces as r(u,v) = (x(u,v), y(u,v), z(u,v)). Learn coordinate patches, regular surfaces, and non-degenerate parameterizations.
Define tangent vectors rᵤ = ∂r/∂u and rᵥ = ∂r/∂v. Learn tangent plane spanned by {rᵤ, rᵥ} and normal vector N = rᵤ × rᵥ.
Study metric tensor I = E du² + 2F du dv + G dv² where E = rᵤ·rᵤ, F = rᵤ·rᵥ, G = rᵥ·rᵥ. Learn length and angle measurements.
Learn II = L du² + 2M du dv + N dv² where L, M, N involve second derivatives. Understand how this measures bending of the surface.
Find principal curvatures κ₁, κ₂ as eigenvalues of the shape operator S = I⁻¹II. Learn principal directions as corresponding eigenvectors.
Define mean curvature H = (κ₁ + κ₂)/2 and Gaussian curvature K = κ₁κ₂. Understand geometric interpretation and computational formulas.
Study curves on surfaces with zero geodesic curvature. Learn geodesic equations and that geodesics locally minimize distance on surfaces.
Compute curvatures for sphere (K = 1/R², H = 1/R), cylinder (K = 0, H = 1/2R), and torus. Understand different geometric behaviors.
Study properties of surfaces that depend only on their intrinsic metric, not their embedding in ℝ³.
Learn Gauss's remarkable theorem: Gaussian curvature depends only on the first fundamental form, not on how the surface sits in space.
Compute Christoffel symbols Γᵢⱼᵏ from the metric tensor. Understand their role in describing how coordinate bases change from point to point.
Learn covariant derivative ∇ as the intrinsic derivative operator on surfaces. Understand how it measures rate of change along the surface.
Study parallel transport of vectors along curves on surfaces. Learn that parallel transport generally depends on the path taken.
Define geodesic curvature κₘ measuring how much a curve deviates from being a geodesic. Learn its relationship to normal curvature.
Learn ∮κₘds + ∬K dA = 2πχ(S) relating curvature to topology. Understand this fundamental connection between geometry and topology.
Study maps that preserve distances and angles. Learn local isometry theorem and understand when surfaces can be "unrolled" without distortion.
Study surfaces with zero Gaussian curvature: cylinders, cones, and tangent developables. Learn their special geometric properties.
Learn the foundations of smooth manifold theory, generalizing curves and surfaces to arbitrary dimensions.
Learn n-dimensional manifolds as spaces locally homeomorphic to ℝⁿ. Understand charts, atlases, and compatibility conditions.
Define smooth manifolds using smoothly compatible coordinate charts. Learn maximal atlases and equivalent smooth structures.
Master coordinate charts φ: U → ℝⁿ and transition maps. Understand how smoothness is defined using local coordinates.
Define smooth maps f: M → N between manifolds. Learn that smoothness is checked using coordinate representations.
Define tangent space TₚM as space of tangent vectors at p. Learn equivalence of geometric, algebraic, and kinematic definitions.
Study differential dfₚ: TₚM →