I took my first course in differential geometry as a graduate student. I got an A, but I didn't learn much. Many of my colleagues, including several non-mathematicians with a desire to learn the subject, have reported similar experiences.

Why should this be the case? I believe there are two reasons. First, differential geometry — like calculus — tends to be taught as a branch of analysis, not geometry. Everything is a map between suitable spaces: Curves and surfaces are parameterized; manifolds are covered with coordinate charts; tensors act on vectors; and so on. This approach may be good mathematics, but it is not very enlightening for beginners. Second, too much attention is given to setting up a general formalism, the tensor calculus. Differential geometry has been jokingly described as the study of those objects which are invariant under changes in notation, but this description is a shockingly accurate summary of the frustrations numerous students experience when trying to master the material.

This book represents my attempt to do something different. The goal is to learn just enough differential geometry to be able to learn the basics of general relativity, although relativity itself is the topic of another book. Furthermore, the book is aimed not only at graduate students, but also at advanced undergraduates, not only in mathematics, but also in physics.

These goals lead to several key choices. This book is about differential forms, not tensors, which are mentioned only in passing. We work almost exclusively in an orthonormal basis, both because it simplifies computations and because it avoids mistaking coordinate singularities for physical ones. And we are quite casual about concepts such as coordinate charts, topological constraints, and differentiability. Instead, we simply assume that our various objects are sufficiently well-behaved to permit the desired operations. The details can, and in my opinion should, come later.

This framework nonetheless allows us to recover many standard, beautiful results in $\RR^3$. We derive formulas for the Laplacian in orthogonal coordinates. We discuss — but do not prove — Stokes' Theorem. We derive both Gauss's Theorema Egregium about intrinsic curvature and the Gauss-Bonnet Theorem relating geometry to topology. But we also go well beyond $\RR^3$. We discuss the Cartan structure equations and the existence of a unique Levi-Civita connection. And we are especially careful not to restrict ourselves to Euclidean signature, using Minkowski space as a key example.

Yes, there is still much formalism to master. Furthermore, this classical approach is no longer standard — and certainly not as an introduction to relativity. I hope to have presented a coherent path to relativity for the interested reader, with some interesting stops along the way.