Download e-book for iPad: An Introduction to Manifolds (2nd Edition) (Universitext) by Loring W. Tu

By Loring W. Tu

ISBN-10: 1441974008

ISBN-13: 9781441974006

Manifolds, the higher-dimensional analogues of tender curves and surfaces, are basic gadgets in sleek arithmetic. Combining facets of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, basic relativity, and quantum box conception. during this streamlined advent to the topic, the speculation of manifolds is gifted with the purpose of assisting the reader in attaining a swift mastery of the fundamental subject matters. by way of the top of the e-book the reader could be capable of compute, at the very least 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 helpful 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 workouts further. This paintings can be used as a textbook for a one-semester graduate or complicated undergraduate direction, in addition to by way of scholars engaged in self-study. The needful point-set topology is incorporated in an appendix of twenty-five pages; different appendices evaluation proof from actual research and linear algebra. tricks and ideas are supplied to some of the routines and difficulties. Requiring simply minimum undergraduate necessities, "An advent to Manifolds" can also be a superb beginning for the author's booklet with Raoul Bott, "Differential kinds in Algebraic Topology."

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Additional resources for An Introduction to Manifolds (2nd Edition) (Universitext)

Example text

K, the number of transpositions required to move j to its natural position is the same as the number of inversions ending in j. e, the ordered list 1, 2, . . , k, from σ (1), σ (2), . . , σ (k) by multiplying σ by as many transpositions as the total number of inversions in σ . Therefore, sgn(σ ) = (−1)# inversions in σ . 3 Multilinear Functions Denote by V k = V × · · · × V the Cartesian product of k copies of a real vector space V . A function f : V k → R is k-linear if it is linear in each of its k arguments: f (.

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.

4) v = ∑ vi i . ∂x p The vector space D p (Rn ) of derivations at p, although not as geometric as arrows, turns out to be more suitable for generalization to manifolds. 4 Vector Fields A vector field X on an open subset U of Rn is a function that assigns to each point p in U a tangent vector X p in Tp (Rn ). Since Tp (Rn ) has basis {∂ /∂ xi | p }, the vector X p is a linear combination X p = ∑ ai (p) ∂ ∂ xi , p p ∈ U, ai (p) ∈ R. Omitting p, we may write X = ∑ ai ∂ /∂ xi , where the ai are now functions on U.

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