Smooth Manifolds

Abstract

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This book is about smooth manifolds. In the simplest terms, these are spaces that locally look like some Euclidean space ℝⁿ, and on which one can do calculus. The most familiar examples, aside from Euclidean spaces themselves, are smooth plane curves such as circles and parabolas, and smooth surfaces such as spheres, tori, paraboloids, ellipsoids, and hyperboloids. Higher-dimensional examples include the set of unit vectors in ℝⁿ⁺¹ (the n-sphere) and graphs of smooth maps between Euclidean spaces.