By Loring W. Tu

Manifolds, the higher-dimensional analogues of gentle curves and surfaces, are basic gadgets in sleek arithmetic. Combining points of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, common relativity, and quantum box conception. during this streamlined advent to the topic, the idea of manifolds is gifted with the purpose of aiding the reader in achieving a swift mastery of the fundamental subject matters. by means of the tip of the booklet the reader could be capable of compute, no less than for easy areas, probably the most simple topological invariants of a manifold, its de Rham cohomology. alongside the best way the reader acquires the information and abilities beneficial for extra learn of geometry and topology. the second one variation comprises fifty pages of latest fabric. Many passages were rewritten, proofs simplified, and new examples and workouts extra. This paintings can be utilized as a textbook for a one-semester graduate or complex undergraduate direction, in addition to by way of scholars engaged in self-study. The considered necessary point-set topology is integrated in an appendix of twenty-five pages; different appendices assessment proof from actual research and linear algebra. tricks and strategies are supplied to a number of the routines and difficulties. Requiring basically minimum undergraduate necessities, "An advent to Manifolds" is additionally a great origin for the author's e-book with Raoul Bott, "Differential types in Algebraic Topology."

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

**Sample text**

Show that a basis for the space Lk (V ) of k-linear functions on V is {α i1 ⊗ · · · ⊗ α ik } for all multi-indices (i1 , . . , ik ) (not just the strictly ascending multi-indices as for Ak (L)). In particular, this shows that dim Lk (V ) = nk . 4. A characterization of alternating k-tensors Let f be a k-tensor on a vector space V . Prove that f is alternating if and only if f changes sign whenever two successive arguments are interchanged: f (. . , vi+1 , vi , . ) = − f (. . , vi , vi+1 , . ) for i = 1, .

N be the dual basis for V ∨ . Introduce the multi-index notation I = (i1 , . . , ik ) and write eI for (ei1 , . . , eik ) and α I for α i1 ∧ · · · ∧ α ik . A k-linear function f on V is completely determined by its values on all k-tuples (ei1 , . . , eik ). If f is alternating, then it is completely determined by its values on (ei1 , . . , eik ) with 1 ≤ i1 < · · · < ik ≤ n; that is, it suffices to consider eI with I in strictly ascending order. 28. Let e1 , . . , en be a basis for a vector space V and let α 1 , .

3) ∂ ∂ xi . p Since Dv is clearly linear in v, the map φ is a linear map of vector spaces. 1. If D is a point-derivation of C∞ p , then D(c) = 0 for any constant function c. Proof. Since we do not know whether every derivation at p is a directional derivative, we need to prove this lemma using only the defining properties of a derivation at p. By R-linearity, D(c) = cD(1). So it suffices to prove that D(1) = 0. 2), D(1) = D(1 · 1) = D(1) · 1 + 1 · D(1) = 2D(1). Subtracting D(1) from both sides gives 0 = D(1).