By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Booklet by way of Michael Spivak, Spivak, Michael

**Read or Download A Comprehensive Introduction to Differential Geometry Volume 2, Third Edition PDF**

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**Additional info for A Comprehensive Introduction to Differential Geometry Volume 2, Third Edition**

**Example text**

Thus 9 is not smooth on U. Now let M be a topological manifold with boundary. Just as in the manifold case, a smooth structure for M is defined to be a maximal smooth atlas-a collection of charts whose domains cover M and whose transition maps (and their inverses) are smooth in the sense just described. With such a structure, M is called a smooth manifold with boundary. A point p E M is called a boundary point if its image under some smooth chart is in 8lHln, and an interior point if its image under some smooth chart is in lnt lHln.

Thus the collection of charts {(Ui±, 'Pt)} is a smooth atlas, and so defines a smooth structure on §n. We call this its standard smooth structure. 21 (Projective Spaces). 3. We will show that the coordinate charts (Ui , 'Pi) constructed in that example are all smoothly compatible. Assuming for convenience that i > j, it is straightforward to compute that which is a diffeomorphism from 'Pi(Ui n Uj ) to 'Pj(Ui n Uj ). 22 (Smooth Product Manifolds). If M 1 , ... , Mk are smooth manifolds of dimensions nl, ...

It is surjective but not injective, because its kernel consists of the complex numbers of the form 27rik, where k is an integer. (d) The map e: JR -t §l defined by e(t) = e 27rit is a Lie group homomorphism whose kernel is the set Z of integers. Similarly the map en: JRn -t Tn defined by en(tb ... ,tn) = (e27ritl, ... ,e27ritn) is a Lie group homomorphism whose kernel is zn. 40 2. 4. A covering map. (e) The determinant function det: GL(n,lR) -+ IR* is smooth because det A is a polynomial in the matrix entries of A.

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