By Loring W. Tu

Manifolds, the higher-dimensional analogs of delicate curves and surfaces, are primary items in glossy arithmetic. Combining features 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 aiding the reader in achieving a quick mastery of the fundamental issues. through the tip of the booklet the reader might 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 helpful for additional examine of geometry and topology. The needful point-set topology is integrated in an appendix of twenty pages; different appendices evaluation evidence from actual research and linear algebra. tricks and recommendations are supplied to a few of the workouts and problems.

This paintings can be used because the textual content for a one-semester graduate or complex undergraduate direction, in addition to through scholars engaged in self-study. Requiring in basic terms minimum undergraduate prerequisites, *An Introduction to Manifolds* can be a superb beginning for Springer GTM eighty two, *Differential kinds in Algebraic Topology.*

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**Example text**

2. The transition function ψ ◦ φ −1 is defined on φ(U ∩ V ). Since we are interested only in C ∞ -compatible charts, we often omit to mention and speak simply of compatible charts. 7. , such that M = α Uα . Although the C ∞ compatibility of charts is clearly reflexive and symmetric, it is not transitive. The reason is as follows. Suppose (U1 , φ1 ) is C ∞ -compatible with (U2 , φ2 ), and (U2 , φ2 ) is C ∞ -compatible with (U3 , φ3 ). Note that the three coordinate functions are simultaneously defined only on the triple intersection U123 .

Vk ). * 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 , . . , α k are linearly independent in the dual space V ∗ . * Exterior multiplication Let α be a nonzero 1-covector and ω a k-covector on a finite-dimensional vector space V . Show that α ∧ ω = 0 if and only if ω = α ∧ τ for some (k − 1)-covector τ on V . 11. Pullback of a k-covector For any linear map L : V − → W of vector spaces and any positive integer k, there is a pullback map L∗ : Ak (W ) − → Ak (V ) defined by L∗ (f )(v1 , .

N } for V ∗ is said to be dual to the basis {e1 , . . , en } for V . 2. The dual space V ∗ of a finite-dimensional vector space V has the same dimension as V . 3 (Coordinate functions). With respect to a basis e1 , . . , en for a vector space V , every v ∈ V can be written uniquely as a linear combination v = bi (v)ei , where bi (v) ∈ R. Let α 1 , . . , α n be the basis of V ∗ dual to e1 , . . , en . Then ⎛ ⎞ α i (v) = α i ⎝ bj (v)ej ⎠ = j bj (v)α i (ej ) = j bj (v)δji = bi (v). j Thus, the set of coordinate functions b1 , .