Learning Path — Differential & Riemannian Geometry
Suggested preparation for Differential & Riemannian Geometry. Choose an entry point that fits your preparation.
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Entry / bridge route · Additional preparation for research reading · All paths
Entry / bridge route
Proofs & Mathematical Writing
Read quantified statements; write direct, contrapositive, contradiction and induction arguments.
Book of Proof — Richard Hammack. Direct open PDF
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You can explain the quantifiers in a statement and write these forms of proof without relying on a worked solution.
Linear Algebra
Work with vector spaces, bases, linear maps, kernels, rank and eigenvalues.
Linear Algebra — Jim Hefferon. Open online text / archive
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You can prove rank–nullity, change bases and distinguish diagonalizable maps from general linear maps.
Calculus
Use limits, derivatives, integrals and series; distinguish computation from justification.
Calculus, Volume 1 — Gilbert Strang and Edwin 'Jed' Herman. Direct open PDF
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You can solve representative differentiation, integration and convergence exercises and explain the hypotheses used.
Point-Set Topology
Understand open sets, continuity, products, quotients and compactness.
Topology Without Tears — Sidney A. Morris. Direct open PDF
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You can prove statements using these definitions and produce basic separation or compactness counterexamples.
Differential & Riemannian Geometry
Work with smooth manifolds, tangent spaces, differential forms and metrics.
What Is Differential Geometry? Curves and Surfaces — Anton Petrunin and Sergio Zamora Barrera. Direct open PDF
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You can change coordinates, use the tangent map and compute exterior derivatives; curvature and geometric-analysis gaps may remain.
Research reading: additional tools
Choose according to the particular text, not as an obligation to complete every foundation again.