Differential Topology
by Victor Guillemin, Alan Pollack · Victor Guillemin, Alan Pollack
Concise undergraduate treatment of manifolds through transversality, intersection theory, degree, and the Poincaré-Hopf theorem, developed for subsets of Euclidean space. Working the exercises leaves a reader able to argue geometrically about smooth maps and their singularities.
This link may earn us a small commission at no extra cost to you. Affiliate disclosure
More resources on Manifolds
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Manifold
Comprehensive reference with category-theoretic perspective, often linked in diff geom discussions
Introduction to Smooth Manifolds
Graduate textbook building smooth manifold theory from topological foundations through tangent bundles, vector fields, differential forms, integration, and de Rham cohomology. Readers finish able to work comfortably with charts, submersions, Lie derivatives, and Stokes' theorem on manifolds.
Topology from the Differentiable Viewpoint
Short lecture-note volume reaching Sard's theorem, degree theory, the Hopf theorem, and Pontryagin's framed-cobordism construction in under eighty pages. It shows how smooth-manifold arguments settle topological questions that combinatorial methods handle far less directly.
A Comprehensive Introduction to Differential Geometry
Multi-volume treatise developing manifolds, connections, curvature, and Riemannian geometry alongside the historical papers of Gauss and Riemann. Reading it supplies both the modern formalism and the reasons each definition took the shape it did.
nLab
Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.