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I am currently a high school student interested in a research career in physics. I have self taught myself single variable calculus and elementary physics upto the level of IPHO . And I am comfortable with euclidean geometry upto the point of some basic theorems like Pythagoras or Thales and the geometry in Cartesian coordinates and use of vectors.

I have read at a lot of places that somehow geometry is very much essential for doing physics? Now I don't know in what context, as in do you ever need the complex geometrical theorems that are upto the level of IMO, in physics or are you just good knowing simple theorems and their proofs and trigonometric relations and similarity etc.

Plus what other forms of geometry are essential for doing physics?

Is linear algebra a form of geometry?

What next step should I take to dive into geometry that will help me in my physics career for making an early start? Non-euclidean? And what is it about the geometry names, hyperbolic, elliptical? I can understand if someone tells me that plane geometry / 3d geometry, but what is the essence of hyperbolic/elliptical geometry?

And what are non euclidean geometries and topologies?

Is there a theoretical minimum that you need in all forms of geometries to do physics? I want to learn the math now properly so that I get used to its machinery and later on, don't have to struggle with it later, especially geometry, since the names of these geometries intrigue me .

Is there any need for learning euclidean geometry, upto the levels of Geometry Revisited by Coxeter?

Also I'd be grateful, if some book recommendations are also provided.

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    $\begingroup$ Hi Andrew. Welcome to Physics.SE. First, Please have a look on both the tag definitions. And, please don't ask too many questions on a single post. You can ask them as a new questions ;-) $\endgroup$ – Waffle's Crazy Peanut Apr 20 '13 at 17:03
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    $\begingroup$ That's a whole lot of questions, many of which are rather open-ended. Just a short response: (1) Contest-style physics (or math) is quite unrelated to research; you'll know if you really like research-style once you're taking advanced classes in college. (2) The geometry required by all physics is basic trig - nothing more fancy or complicated. (3) Certain branches of physics (GR, strings) use more advanced "geometry," but this is something entirely different (differential or algebraic), and knowing more Euclidean geometry won't necessarily help with these. $\endgroup$ – user10851 Apr 20 '13 at 17:08
  • $\begingroup$ @ChrisWhite So , what should I do next in learning geometry ? $\endgroup$ – Andrew Tom Apr 20 '13 at 17:13
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/234/2451 and links therein. $\endgroup$ – Qmechanic Apr 20 '13 at 17:15
  • $\begingroup$ @AndrewTom You're going to need a broader background for the more advanced stuff. Linear algebra is a must for attempting differential geometry, and besides lin. alg. is a must for basically all of physics anyway. Take a step to the side before attempting another step forward. $\endgroup$ – user10851 Apr 20 '13 at 17:18