>>8094070
Lee's smooth manifolds. Spells everything out, hence very suitable for home study.

>>8094080
I just took a look at Lee. i think we aren't studying it so deep. We started with parametric curves in R^2 and R^3 (Tangent, Normal, binormal base). Then we got to elementary parametric surfaces with geodesic curves, first and second fundamental forms, asymptotic lines.
I need something like that for further reading

>>8094096
Sorry I didn't realize they had internet in the 19th century.

>>8094080
>Lee's smooth manifolds.
That isn't really a Geometry book. Only glosses over Riemannian Manifolds and doesn't even define curvature.


For an Undergrad to Beginner grad book I would recommend:

Differential Geometry: Curves, Surfaces, and Manifolds
By Kühnel


For a in-depth grad level treatment :

Riemannian Geometry and Geometric Analysis
By Jost

>>8094100
I understand. I'm not really trying to expend knowledge in geometry, just trying to figure out the basic concept

>>8094103
All geometry is not Riemannian. Why offer such a narrow scope? Also, what do you mean by bla bla is not geometry?

>>8094106
Lee's book isn't Differential Geometry, it is Differential Topology and Analysis.

The distinction between Topology and Geometry is notoriously subtle but Differential Geometry pretty much always implies studying curvature, geodesics, holonomy, etc.

>>8094106
More precisely my question is 'what is geometry according to you?' Clearly your notion of geometry somehow excludes Lie groups, so I would like to know how.

>>8094109
The phrase you're looking for is "G-structures". This way you don't unfairly exclude 80+% of differential geom.

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Accept no substitute

Lee - Introduction to Smooth Manifolds
Milnor - Topology From the Differentiable Viewpoint (very short, great read)
Do Carmo - Riemannian Geometry

>>8094096
Pressley's Elementary Differential Geometry
do Carmo's Differential Geometry of Curves and Surfaces

>>8094070
Lang's Fundamentals of Differential Geometry.

>>8094070
O'Neill's "Semi-Riemannian Geometry With Applications to Relativity". Tries not to overcomplicate things, many examples, great introductory text.

>>8094103
>Jost
This one is great.

>>8094109
Almost no professor you talk to will classify Lee as a topology book, though. It is true that there is no geometry in it (I, like you, consider "geometry" to be a manifold with more than just a smooth structure). But Lee is so obviously meant to prepare the reader for geometry that I think it's fair to just label it as a differential geometry book. It's VERY obviously focused on getting the reader into other areas of differential geometry, and not as focused on differential topological questions (compare to, say, Hirsch or Kosinski's books which I think are more obviously topological, or even Gullemin Pollack).

>>8094111
I don't know ANYONE who includes the study of Lie groups as differential geometry. I only know people who consider it a fundamental and useful tool in differential geometry. Just like a complex geometer MUST know a ton about characteristic classes, but charactristic classes certainly aren't a part complex geometry.

Frankel's "Geometry of Physics"

Differential Geometry by Shifrin is pretty good stuff.