I want a quick insight in differential geometry but it is hard to start directly although i have done courses in calculus and basic algebra .is it necessary to get through point set topology and algebraic topology before reading differential geometry .
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1Here are some thoughts: if you want to do "classical differential geometry", i.e. pieces of surfaces in $\mathbb R^3,$ you should be fine with calculus in $\mathbb R^3,$ i.e. multidimensional calculus. If you took a calculus course, you should have seen some point set topology anyway, e.g. neighborhoods of points. I suggest you just try some books on classical differential geometry. – jflipp Apr 13 '15 at 08:42
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list some books on differential geometry i had read multidimensional calculus but having little knowldege of point set topology please refer easily readable text .i am reading lee introduction to manifolds. – Nebo Alex Apr 13 '15 at 13:38
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Have a look at this book. – jflipp Apr 13 '15 at 14:33
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Algebraic topology is not used heavily in introductory differential geometry (though it does become very important later on). However to have anything beyond a calculus-like differential geometry course, you will need to know point set topology really well as most of the really critical ideas will require it (Urysohn's lemma shows up a lot either explicitly or implicitly for example). – Cameron Williams Apr 13 '15 at 22:46
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There are plenty of differential geometry books which are basically self-contained when it comes to point set topology. I recommend editing your post to explain your motivation for reading about differential geometry in the first place. This will help people tailor their advice to your particular situation. Most books will require you to know the most basic definitions, e.g. open sets, continuous maps, homeomorphisms (all of which you can probably understand for metric spaces like $\mathbb{R}^n$). Since your current question is rather general, here are a few standard (but very different) books:
Manfredo do Carmo's Differential Geometry of Curves and Surfaces: See the appendix to chapter 2 for the point-set topology and multivariable calculus review.
Barrett O'Neill's Elementary Differential Geometry: This book is less "mature" than do Carmo's text, but it has the distinct advantage of emphasizing pullbacks/pushforwards and differential forms. It contains all the point set topology you need.
John Milnor's Topology from the Differential Viewpoint: Brief and beautiful. It's really about manifolds and differential topology, so you should stick to the first two options if you are more concerned with more geometric ideas like curvature and geodesics on Riemannian manifolds.
John Lee's Introduction to Smooth Manifolds: Very self-contained, includes appendices on point-set topology, linear algebra, differential equations, et cetera. However, probably more useful as a reference than a guide for developing "quick intuition". Again, it is less geometric than the first two texts.
Also, check out the suggestions here for more ideas:
