Differential topology and Tu's "An Introduction to Manifolds"

Question

Very soft question, is the content of Tu's : "An Introduction to Manifolds" what is usually covered in "differential topology"?

Answer 1

No, this is a standard course on differentiable manifolds. (I do not know the book personally. The table of contents seems to be missing general vector bundles and, more important, the Frobenius Theorem.) Differential topology (as you can see from, e.g., Guillemin and Pollack's undergraduate book and Hirsch's graduate text) usually incorporates notions of transversality, intersection numbers, and their applications, say, to Euler classes of oriented vector bundles.

That said, the terms differential geometry and differential topology get bandied about quite generally to refer to standard material on differentiable manifolds —whether (in the former case) curvature and connections on bundles or (in the latter case) transversality and intersection theory are present in the courses or not.