By Loring W. Tu

Manifolds, the higher-dimensional analogues of soft curves and surfaces, are primary items in glossy arithmetic. Combining points of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, basic relativity, and quantum box idea. during this streamlined creation to the topic, the idea of manifolds is gifted with the purpose of supporting the reader in achieving a speedy mastery of the fundamental subject matters. through the top of the ebook the reader will be capable of compute, no less than for easy areas, essentially the most easy topological invariants of a manifold, its de Rham cohomology. alongside the way in which the reader acquires the data and talents important for additional examine of geometry and topology. the second one version includes fifty pages of recent fabric. Many passages were rewritten, proofs simplified, and new examples and routines additional. This paintings can be used as a textbook for a one-semester graduate or complicated undergraduate path, in addition to by means of scholars engaged in self-study. The considered necessary point-set topology is integrated in an appendix of twenty-five pages; different appendices evaluation evidence from actual research and linear algebra. tricks and suggestions are supplied to some of the workouts and difficulties. Requiring purely minimum undergraduate necessities, "An creation to Manifolds" can be an outstanding starting place for the author's e-book with Raoul Bott, "Differential types in Algebraic Topology."

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**Example text**

Suppose two sets of vectors u1 , . . , uk and v1 , . . , vk in V are related by k uj = ∑ aij vi , j = 1, . . , k, i=1 for a k × k matrix A = [aij ]. Show that f (u1 , . . , uk ) = (det A) f (v1 , . . , vk ). 9. Vanishing of a covector of top degree Let V be a vector space of dimension n. Prove that if an n-covector ω vanishes on a basis e1 , . . , en for V , then ω is the zero covector on V . * Linear independence of covectors Let α 1 , . . , α k be 1-covectors on a vector space V . Show that α 1 ∧ · · · ∧ α k = 0 if and only if α 1 , .

It took the genius of Hermann Grassmann, a nineteenth-century German mathematician, linguist, and high-school teacher, to recognize the importance of multicovectors. He constructed a vast edifice based on multicovectors, now called the exterior algebra, that generalizes parts of vector calculus from R3 to Rn . 6). Grassmann’s work was little appreciated in his lifetime. D. thesis rejected, because the leading mathematicians of his day such as M¨obius and Kummer failed to understand his work. It was only at the turn of the twentiHermann Grassmann eth century, in the hands of the great differential ge(1809–1877) ´ Cartan (1869–1951), that Grassmann’s ometer Elie exterior algebra found its just recognition as the algebraic basis of the theory of differential forms.

Remark. In some books the notation for σ f is f σ . In that notation, ( f σ )τ = f τσ , not f στ . 5 The Symmetrizing and Alternating Operators Given any k-linear function f on a vector space V , there is a way to make a symmetric k-linear function S f from it: (S f )(v1 , . . , vk ) = ∑ f vσ (1) , . . , vσ (k) σ ∈Sk or, in our new shorthand, Sf = ∑ σ f. σ ∈Sk Similarly, there is a way to make an alternating k-linear function from f . Define Af = ∑ (sgn σ )σ f . 12. If f is a k-linear function on a vector space V , then (i) the k-linear function S f is symmetric, and (ii) the k-linear function A f is alternating.