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.