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I recently started taking an interest in General Relativity, however I am well aware that I simply have no knowledge of the mathematics behind it, especially Riemannian geometry.

That being said, I wish to learn as much as possible about Riemannian geometry, but am a year or so away from even being an undergraduate therefore my knowledge is severely lacking.

I have had a look at some related questions asking for book recommendations and have even (tried to) read some of them, but have found that they expect a far stronger mathematical background than my own, hence my question:

Assuming an undergraduate student in their first or second year at university, what topics would they need a thorough understanding of before they read any introductions to Riemannian geometry?

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    You'll want to know at a minimum: real analysis, linear algebra, general topology, a little algebraic topology, and some smooth manifold theory. – 2017-02-11
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    From my experience as an (ex-)physics student (surrounded by many other physics students), I can tell you that you can understand basic general relativity at some level *without* much of the background that one would need to rigorously understand the underlying mathematical structures. Whether that's the way you want, or *should* want, to go is another issue, but many physics students do precisely that. Once you've got a good grip of classical electrodynamics (e.g. all of Griffiths' book) and know some special relativity, you're probably ready to give it a shot. Try Carroll's book. – 2017-02-11
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    A solid understanding of vector calculus is very important, in particular how Green's Theorem, Stokes' Theorem, and the Divergence Theorem are re-expressed in the language of differential form, and (as mentioned by @Danu) how Maxwell's equations are re-expressed in that language. – 2017-02-11
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    @LeeMosher In fact, using that the exterior derivative is equal to the antisymmetrized covariant derivative, physics textbooks often get away without even talking about forms much at all---but I agree it is not something that is nice to skip. – 2017-02-11

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