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A,. 49) closely. 46). 7 Bibliography The following textbooks treat their subjects at roughly the level I have assumed for this book. The list is intended t o be a guide t s the level of the material; it is certainly not a comprehensive list for background reading. Elementary calculus : G, B. , 1960). Mechanics: K. R. , 19 5 3); or H. , 1950); or L. Landau & E. M. Lifshitz, Mechanics (Pergamon, New York, 1960). Thermodynamics: M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill, New York, 1957); or E.

11 A deeper look at fiber bundles 39 nowhere zero. But in fact there is no C" vector field on S2 which is nowhere zero. This is a consequence of the famous but difficult jked-point theorem of the sphere, that every 1-1 map (diffeomorphism) of S2 onto itself leaves at least one point of S2 fixed. 1 below. Therefore T S ~does not have a global product structure. This is an example in which the bundle is nontrivial because of the topology of the base manifold, S2. (ii) The second example shows that one can actually make a bundle nontrivial even if the base space allows a trivial bundle.

The sphere as a manifold One of the simplest examples of a manifold, which illustrates the importance of allowing for more than one chart, is the sphere. ) Consider the two-sphere ~ (called s2), the set of points in lZ3 for which (xl )2 (x2)2 ( x ~ =) const. 4). This shows that the map involved certainly will not preserve lengths or angles. As a specific example of a map, consider the usual spherical coordinates, with 0 -xl and # x 2 . 5. But there are some funny features here. First, the map breaks down at the pole 0 = 0, where one point is 'mapped' t o the whole line x1 = 0 , 0 $ x 2 < 277.