Stage 4 Geometry

Geometry:
Just like number theory, algebraic geometry applies algebra result to study geometry, blah, blah, blah. Algebraic geometry is related to algebra and number theory.

See Geometry & Topology Monographs for further study.

In general:

• Berger M., Pansu P., Berry J.-P. and Saint-Raymond X. Problems in Geometry
• Bredon G.E. Topology and Geometry
• Eves H. Survey of Geometry
• Coxeter H.S.M. Introduction to Geometry
• Hartshorne R. Geometry: Euclid and Beyond - He wrote an excellent text on algebraic geometry. Begin with this, see if you are with him.
• Hilbert D. The Foundations of Geometry (here for FREE!)
• Hilbert D. and Cohn-Vossen S. Geometry and the Imagination
• Hvidsten M. Geometry with Geometry Explorer - See the web for extra material.
• Nikulin V.V. and Shafarevich I.R. Geometries and Groups

Algebraic Geometry:

• Bump D. Algebraic Geometry
• Dolgachev I.V. Introduction to Algebraic Geometry [.pdf] (FREE!)
• Dolgachev I.V. Topics in Classical Algebraic Geometry [.pdf] (FREE!)
• Drozd Y. Introduction to Algebraic Geometry [.pdf] (FREE!)
• Ewald G. Combinatorial Convexity and Algebraic Geometry
• Fulton W. Algebraic Curves: An Introduction to Algebraic Geometry
• Griffiths P. and Harris J. Principles of Algebraic Geometry
• Harris J. Algebraic Geometry: A First Course
• Hartshorne R. Algebraic Geometry (here is a scanned version for FREE!) - Solutions by William Stein, Richard Borcherds, Jinhyun Park and Chris [.pdf].
• Hassett B. Introduction to Algebraic Geometry
• Lang S. Introduction to Algebraic Geometry
• Lefschetz S. Algebraic Geometry
• Milne J.S. Algebraic Geometry (FREE!)
• Reid M. Undergraduate Algebraic Geometry
• Skorobogatov A.N. Algebraic Geometry Lecture Notes [.pdf] (FREE!)
• Smith K.E., Kahanpää L., Kekäläinen P. and Traves W. An Invitation to Algebraic Geometry
• Ueno K. An Introduction to Algebraic Geometry
• Walker R.J. Algebraic Curves

Further Algebraic Geometry:

• Basu S., Pollack R. and Roy M.-F. Algorithms in Real Algebraic Geometry
• Bochnak J., Coste M. and Roy M.-F. Real Algebraic Geometry
• Cox D.A., Little J.B. and O'Shea D.B. Using Algebraic Geometry - Don't forget to check out the typological error here.
• Debarre O. Higher-Dimensional Algebraic Geometry
• Eisenbud D. The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra
• Hirzebruch F. Topological Methods in Algebraic Geometry
• Lang S. Abelian Varieties (here is a scanned version for FREE!)
• Lang S. Introduction to Algebraic and Abelian Functions (here is a scanned version for FREE!)
• Le Potier J. Lectures on Vector Bundles
• Liu Q. Algebraic Geometry and Arithmetic Curves
• Mumford D. Abelian Varieties (here is a scanned version for FREE!)
• Reid M. Chapters on Algebraic Surfaces (FREE!)
• Semple J.G. and Kneebone G.T. Algebraic Projective Geometry
• Ueno K. Algebraic Geometry 1: From Algebraic Varieties to Schemes
  Ueno K. Algebraic Geometry 2: Sheaves and Cohomology
  Ueno K. Algebraic Geometry 3: Further Study of Schemes
• Voisin C. Hodge Theory and Complex Algebraic Geometry Vol 1, 2
• Yang K. Complex Algebraic Geometry

Differential Geometry:
Also consult mathematical physics section below.

• Chern S.S. Chen W.H. and Lam K.S. Lectures on Differential Geometry - Basic topological concspts assumed. A chapter on Finsler geometry is added in the English translation.
  陈省身，陈维桓：《微分几何讲义》，北京大学出版社
• Eisenhart L.P. Transformations of Surfaces
• Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces
• Hwang A.D. Complex Manifolds and Hermitian Differential Geometry [.pdf] (FREE!)
• Ivey T.A. and Landsberg J.M. Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems
• Kennington A.U. Differential Geometry Reconstructed: A Unified Systematic Framework (FREE!) - Comprehensive.
• Kobayashi S. and Nomizu K. Foundations of Differential Geometry Vol 1, 2
• Kobayashi S. Transformation Groups in Differential Geometry
• Lang S. Fundamentals of Differential Geometry
• Lang S. Introduction to Differentiable Manifolds
• Lee J.M. Introduction to Topological Manifolds
• Madore K. An Introduction to Noncommutative Differential Geometry and its Physical Applications - More advanced work.
• Michor P.W. Topics in Differential Geometry [.pdf] (FREE!)
• Montiel S. and Ros A. Curves and Surfaces
• Schoen R. and Yau S.T. Lectures on Differential Geometry
  丘成桐，孙理察：《微分几何讲义》，高等教育出版社
• Spivak M. A Comprehensive Introduction to Differential Geometry 1, 2, 3, 4, 5 - An epic. Read the first two volumes at least. Content can be found here.
• Warner F.W. Foundations of Differentiable Manifolds and Lie Groups (here [.djvu] for FREE!)

Riemannian Geometry:

• Boothby W.M. An Introduction to Differentiable Manifolds and Riemannian Geometry (here [.djvu] for FREE!)
• Chavel I. Riemannian Geometry: A Modern Introduction - Check the correction and addition [.pdf].
• Chern, S.S. and Shen Z. Riemann-Finsler Geometry
• do Carmo M.P. Riemannian Geometry
• Gallot S., Hulin D. and Lafontaine J. Riemannian Geometry
• Gudmundsson S. An Introduction to Riemannian Geometry (FREE!)
• Jost J. Riemannian Geometry and Geometric Analysis
• Lang S. Differential and Riemannian Manifolds
• Moller J.M. Riemannian Geometry [.pdf] (FREE!)
• O'Neill B. Semi-Riemannian Geometry With Applications to Relativity
• Petersen P. Riemannian Geometry
• Wilkins D.R. A Course in Riemannian Geometry [.pdf] (FREE!)

Fractals:

• Barnsley M.F. Fractals Everywhere
• Barnsley M.F. SuperFractals - See the web.
• Edgar G.A. Measure, Topology, and Fractal Geometry - See author's page for errata.
• Falconer K. Fractal Geometry: Mathematical Foundations and Applications
• Mandelbrot B.B. The Fractal Geometry of Nature
• Mörters P. Lecture Notes on Fractal Geometry [.ps] (FREE!)
• Peitgen H.-O., Jürgens H. and Saupe D. Chaos and Fractals: New Frontiers of Sciences (FREE!)

Further readings:

• Audin M. Geometry
• Bobenko A.I. and Suris Y.B. Discrete Differential Geometry: Consistency as Integrability (FREE!)
• Demailly J.-P. Complex Analytic and Differential Geometry [.pdf] (FREE!)
• Gruber P.M. Convex and Discrete Geometry
• Huybrechts D. Complex Geometry: An Introduction
• Jennings G.A. Modern Geometry with Applications
• Matousek J. Lectures on Discrete Geometry