Unit 5 of 10
Topology and differential geometry
About half topology, half curves and surfaces. Manifolds are new in the syllabus and appeared in 2026.
- 7–8questions expected next time
- 7asked in 2025
- 8asked in 2026
- 3ideas asked in both papers
The helix and the sphere answer most geometry questions. Learn their κ, τ and K.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Totally bounded, complete, compact metric spaces
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A metric space is compact exactly when it is complete and totally bounded.
Constant curvature and torsion
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The curve is a circular helix. τ = 0 gives a circle, κ = 0 a line.
Gaussian curvature of a surface
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A sphere of radius r has K = 1/r². For z = f(x, y), K = (f_xx f_yy − f_xy²) / (1 + f_x² + f_y²)², so z = x² + y² has K = 4 at the origin.
Asked in 2026
8 questions. Tap one for the answer and working.
Q34Count something between antipodal points of a sphere using global geometric arguments (1, 2, finite, infinite)D
Answer: Infinite
Antipodal points on S^2 are joined by infinitely many great semicircles (one per meridian).
Sure
Q42Hausdorff second countable regular manifold: number of compatible metrics (0,1,finite,infinite)D
Answer: Infinite
Hausdorff, 2nd countable, regular implies metrizable (Urysohn); if d is compatible so is c*d for any c>0 (also min(d,1) etc.), so infinitely many.
Sure
Q49Curve with k=1, tau=0: dimension of minimal ambient manifold containing itC
Answer: 2
k=1, tau=0 constant: plane circle; minimal ambient dimension 2.
Sure
Q54Every sequence has a Cauchy subsequence: which of compact/totally bounded/complete must holdB
Answer: totally bounded only
Every sequence has a Cauchy subsequence iff X totally bounded; completeness/compactness not implied (e.g. (0,1)).
Sure
Q58Surface z=x^2+y^2: Gaussian curvature at origin via fundamental formsC
Answer: 4
K=(f_xx f_yy - f_xy^2)/(1+f_x^2+f_y^2)^2 = 4 at origin.
Sure
Q62Curve (t,t^2,t^3): curvature at t=0 via Frenet formulasC
Answer: 2
gamma'=(1,0,0), gamma''=(0,2,0) at t=0: k = |gamma' x gamma''|/|gamma'|^3 = 2.
Sure
Q96Compact connected manifold: number of connected componentsD
Connected means exactly 1 component
Sure
Q99Curve with constant nonzero curvature and torsion: circle/line/helix/ellipseC
Constant nonzero curvature and torsion -> circular helix (Frenet equations); circle has zero torsion, line zero curvature
Sure
Asked in 2025
7 questions. Tap one for the answer and working.
Q94Which Tᵢ fails for metric spaces (incorrect statement)Likely bonus marksD
Every metric space is T1, T2, T3 and T4, so all four statements are true; no option is incorrect. Guess D
Matches the official key
Defective: every metric space is T1, T2, T3 and T4, so no statement is incorrect. The official key still gives D.
Q95Totally bounded metric space is separableB
{a,b} and {b,c} are in the family but their intersection {b} is not, so not closed under intersection
Matches the official key
Q96(content garbled in extraction)A
Totally bounded metric space is separable (compact needs completeness too; "reducible" is meaningless)
Matches the official key
Q97Gaussian curvature of sphere radius r (1/r²)D
Sphere of radius r: K = 1/r^2
Matches the official key
Q98Curvature/torsion of r=(a cos u, a sin u, bu)B
Circular helix: torsion = b/(a^2+b^2), curvature a/(a^2+b^2)
Matches the official key
Q99Gaussian curvature of sphere radius aC
K = 1/a^2
Matches the official key
Q100Constant curvature and torsion ⇒ circular helixC
Constant curvature and torsion implies circular helix (standard theorem); "helix"/"cylindrical helix" are also true but weaker
Matches the official key
(C) is the standard answer. (A) cylindrical helix and (B) helix are also true statements (circular helix is a special case), but (C) is the most specific.
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Product and quotient topology
- Tychonoff theorem (asked in 2019)
- Urysohn lemma
- Countability axioms
- First and second fundamental forms, asked directly
- Geodesics
Full syllabus for this unit
Topologies, homeomorphisms, connectedness, compactness, countability, separation axioms, subspace, product and quotient topologies, Tychonoff, Urysohn metrisation; space curves, Serret–Frenet, fundamental forms, Gaussian and principal curvatures, geodesics, manifolds.