By Loring W. Tu

Manifolds, the higher-dimensional analogs of delicate curves and surfaces, are basic gadgets in glossy arithmetic. Combining elements of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, normal relativity, and quantum box theory.

In this streamlined creation to the topic, the idea of manifolds is gifted with the purpose of assisting the reader in achieving a quick mastery of the fundamental subject matters. by way of the top of the ebook the reader will be in a position to compute, a minimum of for easy areas, the most simple topological invariants of a manifold, its de Rham cohomology. alongside the best way the reader acquires the data and talents helpful for extra research of geometry and topology. The considered necessary point-set topology is incorporated in an appendix of twenty pages; different appendices evaluate proof from actual research and linear algebra. tricks and options are supplied to a number of the routines and problems.

This paintings can be used because the textual content for a one-semester graduate or complicated undergraduate path, in addition to via scholars engaged in self-study. Requiring simply minimum undergraduate prerequisites, *An Introduction to Manifolds* is additionally an outstanding origin for Springer GTM eighty two, *Differential varieties in Algebraic Topology.*

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Similarly, ψ ◦ σ −1 is C ∞ on σ (V ∩ W ). 3 Smooth Manifolds An atlas A on a locally Euclidean space is said to be maximal if it is not contained in a larger atlas; in other words, if M is any other atlas containing A, then M = A. 9. A smooth or C ∞ manifold is a topological manifold M together with a maximal atlas. The maximal atlas is also called a differentiable structure on M. A manifold is said to have dimension n if all of its connected components have dimension n. A manifold of dimension n is also called an n-manifold .

16 (The product manifold ). 26). To show that M × N is a manifold, it remains to exhibit an atlas on it. 17 (An atlas for a product manifold). If {(Uα , φα )} and {(Vi , ψi )} are atlases for M and N , respectively, then → Rm+n )} {(Uα × Vi , φα × ψi : Uα × Vi − is an atlas on M × N . Therefore, if M and N are manifolds, then so is M × N . 4 Examples of Smooth Manifolds 53 Proof. 4. 18. 6). Infinite cylinder Torus Fig. 6. Since M × N × P = (M × N ) × P is the successive product of pairs of spaces, if M, N and P are manifolds, then so is M × N × P .

We use the notation [bji ] to denote the matrix whose (i, j )-entry is bji . 28 (Wedge product of 1-covectors). If α 1 , . . , α k are linear functions on a vector space V and v1 , . . , vk ∈ V , then (α 1 ∧ · · · ∧ α k )(v1 , . . , vk ) = det[α i (vj )]. Proof. 4), (α 1 ∧ · · · ∧ α k )(v1 , . . , vk ) = A(α 1 ⊗ · · · ⊗ α k )(v1 , . . , vk ) = (sgn σ )α 1 (vσ (1) ) · · · α k (vσ (k) ) σ ∈Sk = det[α i (vj )]. 10 A Basis for k-Covectors Let e1 , . . , en be a basis for a real vector space V , and let α 1 , .