Differential geometry: frames, compatibility and global obstructions
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Suggested session length: 300 minutes, as an unvalidated planning estimate. This collection contains 11 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt these public mixed problems before revealing worked guidance. They require human mathematical review; saved answers and solution exposure do not establish secure independent examination performance.
Use S=−dN, H=(tr S)/2, and R(X,Y)Z=∇_X∇_YZ−∇_Y∇_XZ−∇_[X,Y]Z throughout. State normal, curve orientation, domain and parameter choices.
Separate computed examples from general proofs. Explicitly identify uses of ODE existence, the inverse function theorem, Wirtinger, Frobenius, Hopf–Rinow, smooth triangulation or Poincaré–Hopf. Work is untimed; the duration is a planning suggestion.
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On a small rectangle prescribe g=4du²+dv² and b=2(1+u)du². Verify positivity, Gauss and Codazzi; construct an immersion by one-dimensional integrals and compute its oriented normal and principal curvatures. Contrast b=2(1+v)du² with the same g. If the compatible immersion is reparameterized by v=−w and the normal is chosen from the new coordinate cross product, how does its second form change? Explain local uniqueness without claiming a global embedding.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For γ(t)=(3 cos t,3 sin t,4t), compute speed, arc length from t=0 to t=2π, curvature and torsion. Give its Frenet frame at t=0. Its planar projection is traversed clockwise instead of counterclockwise: explain the effect on signed planar curvature and why this is different from changing the normal of a surface.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct a unit-speed plane curve with signed curvature κ(s)=2s, initial point (0,0), and initial tangent (1,0), for −1<s<1. Prove uniqueness with those initial data. Explain why the zero at s=0 is harmless here but obstructs the usual Frenet principal normal. For the ellipse (3 cos t,2 sin t), derive a strict lower bound for length from its area using the isoperimetric inequality, and identify the analytic input in a proof of that inequality.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let X(u,v)=(u+v,2u−v,2v), on 0≤u≤2 and 0≤v≤1. Prove regularity, find its oriented unit normal and metric matrix, and compute patch area and the squared norm of the coordinate vector (2,−1). Express its image plane as a regular level set. Explain why exchanging u and v reverses the chart orientation but leaves unsigned area unchanged.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the graph z=(3x²+2xy+3y²)/2, use the upward normal and S=−dN. At the origin find I, II, principal curvatures/directions, K and H, and the normal curvature of v=(cos(π/6),sin(π/6)). Prove why the same normal curvature is obtained by every unit-speed surface curve with this tangent. At (1,0), calculate K and explain why Hessian eigenvalues there are not the principal curvatures.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare the scaled catenoid X(u,v)=(2 cosh v cos u,2 cosh v sin u,2v) and the ruled graph Y(u,v)=(u,v,uv). For the catenoid use N parallel to X_u×X_v and compute I, II, H and K. For Y use the upward normal and compute K and H, showing why being ruled does not imply being minimal or developable. State what H=0 establishes about area variations.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On the sector r>0, 0<θ<π/2 with metric dr²+r²dθ², show that (r,θ)↦(r cos θ,r sin θ) is a local isometry. Derive the nonzero Christoffel symbols. Parallel transport the unit radial vector at θ=0 along the circular arc r=2 to θ=π/3, interpreting endpoints by extending to a slightly larger sector. Give its components in the orthonormal polar frame at the endpoint. Why do nonzero Christoffel symbols here not imply nonzero Gaussian curvature?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On a unit sphere let p=(0,0,1) and v=(3/5,4/5,0). Find the unit-speed geodesic with these initial data and its point at time π/3. Compute its energy on [0,π/3]. Derive the fixed-endpoint first-variation equation for energy and explain, using Gauss’s lemma, why sufficiently short radial geodesics minimize length. Explain why the same sphere geodesic followed to time 3π/2 is not minimizing from p.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On a simply connected planar domain prescribe g=du²+dv² and b=diag(1+v,0). Decide whether these forms can arise from a regular surface, checking both Gauss and Codazzi. Contrast with constant b=diag(1/3,0), giving an explicit local realization. Separately solve j″+9j=0, j(0)=0,j′(0)=1, and interpret its first positive zero on a curvature-9 surface. State the additional theorem needed to turn general compatible forms into a surface.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In the punctured Euclidean plane compare paths from (−2,0) to (3,0): prove their length infimum is five and is not attained, and relate this to metric incompleteness. On the unit sphere take the north cap with boundary colatitude θ=π/4, oriented positively with the outward normal. Compute its area and signed boundary geodesic-curvature integral, and verify Gauss–Bonnet. Specify what Hopf–Rinow would guarantee in a complete connected surface without boundary.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A smooth closed connected oriented surface has genus four. Determine its total Gaussian curvature and decide whether it can have K≥0 everywhere. A continuous tangent field on it has exactly four isolated zeros with three indices 1,−2,−3; find the fourth. Explain why neither conclusion follows unchanged for a surface with boundary. Finally prove that no compact boundaryless immersed surface in R³ is minimal everywhere, stating the identity and integration theorem used.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
On a small rectangle prescribe g=4du²+dv² and b=2(1+u)du². Verify positivity, Gauss and Codazzi; construct an immersion by one-dimensional integrals and compute its oriented normal and principal curvatures. Contrast b=2(1+v)du² with the same g. If the compatible immersion is reparameterized by v=−w and the normal is chosen from the new coordinate cross product, how does its second form change? Explain local uniqueness without claiming a global embedding.
For γ(t)=(3 cos t,3 sin t,4t), compute speed, arc length from t=0 to t=2π, curvature and torsion. Give its Frenet frame at t=0. Its planar projection is traversed clockwise instead of counterclockwise: explain the effect on signed planar curvature and why this is different from changing the normal of a surface.
Construct a unit-speed plane curve with signed curvature κ(s)=2s, initial point (0,0), and initial tangent (1,0), for −1<s<1. Prove uniqueness with those initial data. Explain why the zero at s=0 is harmless here but obstructs the usual Frenet principal normal. For the ellipse (3 cos t,2 sin t), derive a strict lower bound for length from its area using the isoperimetric inequality, and identify the analytic input in a proof of that inequality.
Let X(u,v)=(u+v,2u−v,2v), on 0≤u≤2 and 0≤v≤1. Prove regularity, find its oriented unit normal and metric matrix, and compute patch area and the squared norm of the coordinate vector (2,−1). Express its image plane as a regular level set. Explain why exchanging u and v reverses the chart orientation but leaves unsigned area unchanged.
For the graph z=(3x²+2xy+3y²)/2, use the upward normal and S=−dN. At the origin find I, II, principal curvatures/directions, K and H, and the normal curvature of v=(cos(π/6),sin(π/6)). Prove why the same normal curvature is obtained by every unit-speed surface curve with this tangent. At (1,0), calculate K and explain why Hessian eigenvalues there are not the principal curvatures.
Compare the scaled catenoid X(u,v)=(2 cosh v cos u,2 cosh v sin u,2v) and the ruled graph Y(u,v)=(u,v,uv). For the catenoid use N parallel to X_u×X_v and compute I, II, H and K. For Y use the upward normal and compute K and H, showing why being ruled does not imply being minimal or developable. State what H=0 establishes about area variations.
On the sector r>0, 0<θ<π/2 with metric dr²+r²dθ², show that (r,θ)↦(r cos θ,r sin θ) is a local isometry. Derive the nonzero Christoffel symbols. Parallel transport the unit radial vector at θ=0 along the circular arc r=2 to θ=π/3, interpreting endpoints by extending to a slightly larger sector. Give its components in the orthonormal polar frame at the endpoint. Why do nonzero Christoffel symbols here not imply nonzero Gaussian curvature?
On a unit sphere let p=(0,0,1) and v=(3/5,4/5,0). Find the unit-speed geodesic with these initial data and its point at time π/3. Compute its energy on [0,π/3]. Derive the fixed-endpoint first-variation equation for energy and explain, using Gauss’s lemma, why sufficiently short radial geodesics minimize length. Explain why the same sphere geodesic followed to time 3π/2 is not minimizing from p.
On a simply connected planar domain prescribe g=du²+dv² and b=diag(1+v,0). Decide whether these forms can arise from a regular surface, checking both Gauss and Codazzi. Contrast with constant b=diag(1/3,0), giving an explicit local realization. Separately solve j″+9j=0, j(0)=0,j′(0)=1, and interpret its first positive zero on a curvature-9 surface. State the additional theorem needed to turn general compatible forms into a surface.
In the punctured Euclidean plane compare paths from (−2,0) to (3,0): prove their length infimum is five and is not attained, and relate this to metric incompleteness. On the unit sphere take the north cap with boundary colatitude θ=π/4, oriented positively with the outward normal. Compute its area and signed boundary geodesic-curvature integral, and verify Gauss–Bonnet. Specify what Hopf–Rinow would guarantee in a complete connected surface without boundary.
A smooth closed connected oriented surface has genus four. Determine its total Gaussian curvature and decide whether it can have K≥0 everywhere. A continuous tangent field on it has exactly four isolated zeros with three indices 1,−2,−3; find the fourth. Explain why neither conclusion follows unchanged for a surface with boundary. Finally prove that no compact boundaryless immersed surface in R³ is minimal everywhere, stating the identity and integration theorem used.
