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The book Differential Forms in Algebraic Topology of Bott & Tu gives a nice treatment with the Thom isomorphisms and the Euler classes of vector bundles rank $2$. However, I think their approach to the Euler classes in higher dimensions is quite terrible for novices, at least for me because I do not want to be swamped in a bunch of Čech cohomology stuffs and bicomplexes-arguments.

Therefore, I found another book, Differential Geometry: Connections, Curvature, and Characteristic Classes of Tu in which the author represents Pontryagin, Euler and Chern classes from differential geometry point of view unlike topological methods in Bott & Tu. In particular, he (Tu) constructs the Euler class by mean of curvature matrix and Plaffian but I do not know why he just stops at giving the definition without proving other important axioms characterizing the Euler class as other books do. So I ask for a book doing this job.

Thank you in advance.

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  • $\begingroup$ Two references off the top of my head are Milnor's Characteristic Classes and the other book is chapter 3 of Hatcher's book on vector bundles and K-Theory. Both are available online and are detailed enough for beginners. $\endgroup$
    – shubhankar
    Commented Apr 21, 2020 at 14:55
  • $\begingroup$ @ShubhankarSahai Dear Sahai, both books you mention construct the euler class from the pure algebraic topology while I am seeking for one which uses diff geom. $\endgroup$
    – Alexey Do
    Commented Apr 21, 2020 at 15:03
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    $\begingroup$ In this case you should try the book Geometry of Differential Forms by Morita. Also have a look at the appendix to Milnor's book above. $\endgroup$
    – shubhankar
    Commented Apr 21, 2020 at 15:23

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