Reddit reviews Differential Geometry of Curves and Surfaces

Do Carmo's Differential Geometry of Curves and Surfaces

Gallian's Contemporary Abstract Algebra

I'll second Spivak's two calculus texts. Apostol and Courant are good alternatives if you have some reservations about Spivak.

I'd go with Do Carmo's Differential Geometry of Curves and Surfaces instead of Spivak's five volume sequence.

Manifolds that are Euclidean locally are called Riemann manifolds, but in general, not all manifolds have that property.

My only experience with manifolds is from shape analysis, so I used a Riemann manifold to measure differences in 2-d closed curves by geodesics. I still don't 'get' them, but you might want to check out the book by Do Carmo on Differential Geometry

http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

From my limited understanding, a Riemann manifold is a kind of generic space to compare curves in other spaces that might not normally be comparable because of curvature. Like if you want to compare a line in Euclidean coordinates to a 'line' in spherical coordinates, you'd transform each curve using the xyz or R,theta, phi, plop them on a manifold and calculate the difference using an inner product on the tangent space.

Differential geometry track. I'll try to link to where a preview is available. Books are listed in something like an order of perceived difficulty. Check Amazon for reviews.

Calculus

Thompson, Calculus Made Easy. Probably a good first text, well suited for self-study but doesn't cover as much as the next two and the problems are generally much simpler. Legally available for free online.

Stewart, Calculus. Really common in college courses, a great book overall. I should also note that there is a "Stewart lite" called Calculus: Early Transcendentals, but you're better off with regular Stewart. Huh, it looks like there's a new series called Calculus: Concepts and Contexts which may be a good substitute for regular Stewart. Dunno.

Spivak, Calculus. More difficult, probably better than Stewart in some sense.

Linear Algebra

Poole, Linear Algebra. I haven't read this one but it has great reviews so I might as well include it.

Strang, Introduction to Linear Algebra. I think the Amazon reviews summarize how I feel about this book. Good for self-study.

Differential Geometry

Pressley, Elementary Differential Geometry. Great text covering curves and surfaces. Used this one in my undergrad course.

Do Carmo, Differential Geometry of Curves and Surfaces. Probably better left for a second course, but this one is the standard (for good reason).

Lee, Riemannian Manifolds: An Introduction to Curvature. After you've got a grasp on two and three dimensions, take a look at this. A great text on differential geometry on manifolds of arbitrary dimension.

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Start with calculus, studying all the single-variable stuff. After that, you can either switch to linera algebra before doing multivariable calculus or do multivariable calculus before doing linear algebra. I'd probably stick with calculus. Pay attention to what you learn about vectors along the way. When you're ready, jump into differential geometry.

Hopefully someone can give you a good track for the other geometric subjects.

Before it was re-published by Dover, Differential Geometry of Curves and Surfaces was green too; now it's blue, and the only green book by do Carmo still in publication is Riemannian Geometry.

There are two classes you might have slept through with that name. The "classical" differential geometry of curves and surfaces (I think the standard is do Carmo), or a class on Riemannian geometry (I can recommend Lee).