Rudin type book for differential geometry and algebra

[martin_blckrs]

I'm currently taking graduate courses on differential geometry and algebra. What books are closest to the style of Rudin for these areas (i.e. rigorous, developing the theory in apropriate generality and being elegant at the same time).

For Algebra, I guess Lang is the bible, but what else is there? I would especially welcome some book that covers some advanced linear algebra, like symplectic and complex structures, matrix groups, etc.

For Differential Geometry, I have already tried a lot of books, but none of them really fit my needs. Kobayashi and Nomizu is almost unreadable for me and it deals mostly with bundles. On the other hand, there's Lee's Introduction to smooth manifolds, which has great list of topics, but I find his way of writing ugly. So topic-wise I'm searching for something like Lee, but done in a more elegant way. Is there anything like that?

 

[Landau]

Suggestions:
Algebra: Hungerford's Algebra, Roman's Advanced Linear Algebra
Differential Geometry: Darling's Differential Forms and Connections, Spivak's Comprehensive Introduction to DG, Serge Lang's Fundamentals of DG, Barden and Thomas' An Introduction to Differential Manifolds.

 

[Daverz]

Darling's book is very cool, but the emphasis in the later half is on bundles, if the "connections" part didn't clue you in. Nice for physicists who want to get up to speed on this.

I would suggest checking out https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20, who covers more traditional differential geometry topics but at a more advanced level than the typical undergraduate text.