TL;DR Start here

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I think the classic introduction to the topic is Do Carmo's Riemannian Geometry. One that my colleagues use a lot (and is always taken out of the library, grrr) is Jurgen Jost's Riemannian Geomery and Geometric Analysis this second book is more recent and put out by springer.

There's another set of books that, from what I understand, approaches much more the algebraic aspects of this topic, but I have no experience with it. But I've read a lot of people in that area think it's the bee's knees. This is the 4 volume work by Spivak, A Comprehensive Introduction to Differential Geometry

absolutely no problem. i miss talking about these things and studying these subjects as well, so it's interesting for me.

rosenberg's book is freely available on his website, so you should definitely have a look. i would estimate the book to be on the level of a second year graduate student though, but it should at least give a good idea of things.

i would say that both john lee's and loring tu's introduction to smooth manifold books are the gateway texts. if you have a good understanding of most of the topics there (basically all of tu's book), then you're probably ready to move on, using the "lesser" books as intuition builders. lee's riemannian geometry book would probably be a good place to start with over rosenberg's though, due to it being more straight forward.

there's also riemannian geometry and geometric analysis by jurgen jost that i forgot to mention. it's a german book, so it is rather dense, but it's a good one from what i remember.

lastly, there's a very famous paper called "can one hear the shape of a drum?" by mark kac that you should try to hunt down.

Nakamura is ok. I like Bleecker. The classical reference is Kobayashi and Nomizu. Nakamura is advanced undergrad. Bleecker is masters / post grad and K & N is renowned for both it's rigour and difficulty. From taking a brief look over the notes you are currently using any of these books would be fine.

I'm a bit surprised that you've had difficulty finding resources. Maybe it's your search terms? Try looking for principal fibre bundles, differential geometry, geometric analysis... etc...

Oh. Speaking of geometric analysis Josh does an ok job of reviewing fibre bundles / connections. There's a little bit of a connection to physics via Yang-Mills.

https://www.amazon.com/Geometry-Topology-Physics-Graduate-Student/dp/0750306068

https://www.amazon.com/Gauge-Theory-Variational-Principles-Physics/dp/0486445461/ref=sr_1_fkmr0_3?s=books&ie=UTF8&qid=1467948564&sr=1-3-fkmr0&keywords=bleecker+guage

https://www.amazon.com/Foundations-Differential-Geometry-Classics-Library/dp/0471157333/ref=sr_1_1?s=books&ie=UTF8&qid=1467948581&sr=1-1&keywords=kobayashi+and+nomizu

https://www.amazon.com/Riemannian-Geometry-Geometric-Analysis-Universitext/dp/3642212972/ref=sr_1_fkmr1_3?s=books&ie=UTF8&qid=1467948599&sr=1-3-fkmr1&keywords=Josh+geometric+analysis