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what is a good book in differential geometry. I currently know calculus, a bit about differential equations, a bit of linear algebra and a bit about tensors. I also know some variational calculus.

Of course what I know won't really help. I've skimmed through some physics sources and mathematical sources and have decided exactly what I want. I currently don't have any.

I'm looking for something clear and intuitive. something that doesn't throw formal definitions in your face without explaning what motivates them. I would like something that explains what lead to certain concepts and theorems. That's what tunred me away from most mathematics books. i don't think it helps a beginner to start a book that opens with "Let a be (insert formal terms here thing)".

I would however like the book to have mathematical rigour. Things, even if somewhat obvious, should be proven and not taken for granted. NOTHING should be taken for granted. This feature turned me away from a great deal of physics texts.

so I would like something that is informal and intuitive but still rigorous. a kind of medium.

PS.: I will admit that I'm a very impatient person. I am working on it though.

Some chapters introducing topology would also be nice.

I personally think that formality these days is killing mathematics. I admit that it is important, but it is also important to see how definitions arise.

Of course what I know won't really help. I've skimmed through some physics sources and mathematical sources and have decided exactly what I want. I currently don't have any.

I'm looking for something clear and intuitive. something that doesn't throw formal definitions in your face without explaning what motivates them. I would like something that explains what lead to certain concepts and theorems. That's what tunred me away from most mathematics books. i don't think it helps a beginner to start a book that opens with "Let a be (insert formal terms here thing)".

I would however like the book to have mathematical rigour. Things, even if somewhat obvious, should be proven and not taken for granted. NOTHING should be taken for granted. This feature turned me away from a great deal of physics texts.

so I would like something that is informal and intuitive but still rigorous. a kind of medium.

PS.: I will admit that I'm a very impatient person. I am working on it though.

Some chapters introducing topology would also be nice.

I personally think that formality these days is killing mathematics. I admit that it is important, but it is also important to see how definitions arise.

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