Courses / Mathematics / Differential geometry of curves and surfaces Mathematics · unit version 1 Differential geometry of curves and surfaces Thirty undergraduate lessons develop curvature, surface metrics and shape, connections, geodesics, Gauss–Codazzi and Gauss–Bonnet through original calculations and proofs. The convention S=−dN is used throughout. Global ODE, integrability, triangulation and topological theorem dependencies are stated explicitly; saved practice is not certification of proof quality or university-course equivalence.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How do you find the tangent plane of a regular parametrized surface? Surface geometry begins with independent tangent directions.
Compare reasoning and find a refresher Differentiate the parametrization in its two coordinates. Their span is the tangent plane when their cross product is nonzero; a zero cross product requires checking regularity or changing the chart.
Review surface parametrizations
Why are eigenvalues of a symmetric matrix real, and what does its Rayleigh quotient measure? The shape operator uses self-adjoint linear algebra to identify principal directions.
Compare reasoning and find a refresher The spectral theorem supplies an orthonormal eigenbasis with real eigenvalues. For a unit vector v, vᵀAv is a weighted average of those eigenvalues and lies between their minimum and maximum.
Review self-adjoint operators
What does the chain rule say when a curve is reparametrized? Geometric quantities must separate the curve from its traversal speed.
Compare reasoning and find a refresher If β(t)=α(s(t)), then β′=α′(s(t))s′(t). Speed changes by |s′|, and an increasing regular change of parameter preserves oriented tangent direction.
Review space curves and motion
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Differential geometry: frames, compatibility and global obstructions
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Lesson details and objectives Separate the geometric trace from its parametrization.
Construct unit-speed coordinates on a regular curve. Prove length invariance under regular reparametrization. Diagnose a singular parametrization of a smooth trace. 6 practice tasks.
Lesson details and objectives Recover local bending from a unit tangent.
Derive signed curvature from the tangent angle. Compute curvature with a non-unit parameter. Track orientation reversal and curvature sign. 6 practice tasks.
Lesson details and objectives Derive the moving frame of a nonstraight space curve.
Prove the Frenet equations where curvature is positive. Compute curvature and torsion from arbitrary parameters. Explain why the frame can fail at zero curvature. 6 practice tasks.
Lesson details and objectives Use an ODE to turn curvature and torsion into a curve.
Construct a plane curve from prescribed signed curvature. Prove orthonormality is preserved by the Frenet ODE. State uniqueness up to orientation-preserving rigid motion. 6 practice tasks.
Lesson details and objectives Connect local curvature to global restrictions.
Express turning number as a tangent winding number. Prove the isoperimetric bound using a stated Wirtinger inequality. Distinguish total signed curvature from total absolute curvature. 6 practice tasks.
Lesson details and objectives Verify that a displayed parametrization describes a surface.
Check rank and local embedding conditions. Use the regular value theorem for level surfaces. Distinguish chart singularity from surface singularity. 6 practice tasks.
Lesson details and objectives Define tangent vectors independently of a chosen chart.
Prove tangent planes are chart-independent. Differentiate surface maps along curves. Track normal orientation under coordinate changes. 6 practice tasks.
Lesson details and objectives Measure tangent lengths and angles with a Gram matrix.
Compute E,F,G from a parametrization. Prove positivity and coordinate transformation of the metric. Use the metric to measure curve speed. 6 practice tasks.
Lesson details and objectives Derive the area element from tangent parallelograms.
Use the Gram determinant to integrate surface area. Prove area invariance under coordinate changes. Avoid counting multiply covered patches as embedded area. 6 practice tasks.
Lesson details and objectives Measure bending relative to a chosen normal.
Derive II from normal acceleration. Compute its coordinate coefficients. Track the effect of reversing orientation. 6 practice tasks.
Lesson details and objectives Diagonalize bending in the tangent metric.
Prove self-adjointness of S. Compute principal curvatures as a generalized eigenproblem. Distinguish Gaussian and mean curvature. 6 practice tasks.
Lesson details and objectives Separate curve curvature from surface bending.
Prove Euler's normal-curvature formula. Derive Meusnier's relation with angle conventions. Classify asymptotic directions from curvature signs. 6 practice tasks.
Lesson details and objectives Interpret curvature as the derivative of the normal map.
Relate the Gauss-map Jacobian to K. Distinguish signed area change from absolute area. Use local invertibility when K is nonzero. 6 practice tasks.
Lesson details and objectives Derive mean curvature and stationarity of area.
Compute K and H for a graph with upward normal. Derive the minimal graph equation by first variation. Distinguish stationary area from global minimality. 6 practice tasks.
Lesson details and objectives Reduce surface calculations to a profile curve.
Compute the metric of an arc-length profile revolution. Derive meridian and parallel principal curvatures. Verify a catenoid has zero mean curvature. 6 practice tasks.
Lesson details and objectives Separate straight generators from intrinsic flatness.
Compute the regularity condition of a ruled surface. Derive its nonpositive Gaussian curvature. Verify the developability determinant criterion. 6 practice tasks.
Lesson details and objectives Determine which geometric properties survive bending.
Verify local isometry by pulling back the metric. Distinguish local isometry from global distance preservation. Compute conformal angles without claiming equal lengths. 6 practice tasks.
Lesson details and objectives Differentiate tangent fields without leaving the surface.
Derive the metric Christoffel formula. Compute a connection in polar coordinates. Explain why nonzero Christoffel symbols do not imply curvature. 6 practice tasks.
Lesson details and objectives Move tangent vectors without intrinsic turning.
Derive the parallel-transport ODE. Prove preservation of inner products. Compute holonomy around a spherical latitude. 6 practice tasks.
Lesson details and objectives Separate intrinsic straightness from ambient straightness.
Derive the coordinate geodesic equation. Prove geodesics have constant speed. Use symmetry to obtain Clairaut's conserved quantity. 6 practice tasks.
Lesson details and objectives Prove the first-variation formula with fixed endpoints.
Derive energy first variation. Distinguish stationary energy from global minimality. Relate length and energy through Cauchy–Schwarz. 6 practice tasks.
Lesson details and objectives Prove a local shortest-path result from radial orthogonality.
Define the exponential map through geodesic initial data. Prove Gauss's radial orthogonality lemma. Use normal coordinates to justify local minimizing segments. 6 practice tasks.
Lesson details and objectives Prove that Gaussian curvature is determined by the metric.
Derive the Gauss equation from ambient flatness. Extract K from the intrinsic curvature tensor. Compute conformal curvature with a fixed sign convention. 6 practice tasks.
Lesson details and objectives Identify the compatibility conditions for a metric and bending form.
Derive Codazzi from the normal ambient commutator. Test whether prescribed forms can come from a surface. State the precise local reconstruction theorem and its integrability dependency. 8 practice tasks.
Lesson details and objectives Derive the linearized equation for neighboring geodesics.
Derive the Jacobi equation from a geodesic variation. Solve normal separation in constant curvature. Distinguish conjugate points from every global cut point. 6 practice tasks.
Lesson details and objectives Separate local geodesic existence from global continuation.
Give an incomplete metric example. Prove compact surfaces are geodesically complete using ODE continuation. State Hopf–Rinow with its global hypotheses. 6 practice tasks.
Lesson details and objectives Split acceleration into surface-normal and tangent-normal parts.
Derive the Darboux decomposition of curve acceleration. Compute signed geodesic curvature on a latitude. Track orientation and corner contributions. 6 practice tasks.
Lesson details and objectives Prove the curvature–boundary identity on a disk patch.
Derive the connection-form curvature identity. Relate boundary turning to geodesic curvature. Prove the local Gauss–Bonnet formula under stated disk hypotheses. 6 practice tasks.
Lesson details and objectives Sum local curvature identities into a topological invariant.
Prove the closed-surface formula by cancellation over triangles. Compute Euler characteristic and total curvature. Extend the accounting to boundaries and corners. 6 practice tasks.
Lesson details and objectives Combine curvature, topology and extrinsic constraints.
Prove a compact boundaryless surface cannot be minimal in R³. Apply the index theorem with explicit hypotheses. Distinguish local reconstruction from global embedding and rigidity. 6 practice tasks.
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