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Geometry-Scenic-Intro

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Comprehensive open source book on basic topology, smooth manifolds, differential geometry, Lie theory, homological algebra, and index theory.

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- algorithm.pdf
- barycentric.png
- Cayley.pdf
- cell_approx.pdf
- collar.png
- coverings_8.pdf
- crosscap.pdf
- CW_SOn.pdf
- cylinder.pdf
- develop.pdf
- develop.pdf_tex
- develop.svg
- development.png
- discont_lift.pdf
- foliations.pdf
- frobenius_1.pdf
- frobenius_2.pdf
- fundamental-domain.png
- geodesic.pdf
- helicoid.pdf
- hep_example.pdf
- higher-homotopy.png
- holonomy.pdf
- hom_of_pairs.pdf
- Hopf.png
- immersion.pdf
- kernel of Phi.pdf
- Klein.pdf
- mobius.png
- null cone.pdf
- oscillator.png
- poincare-hopf.pdf
- poincare.png
- prism.pdf
- projectiveplane.png
- separation-axioms.pdf
- SG1soliton.png
- SG2soliton.png
- SGbreather.png
- sixfold.pdf
- subdivision.pdf
- threefold_cover.pdf
- Top_sin.jpg
- uni_open_set.pdf
- weingarten.pdf
- classical_geom.tex
- diff_geom.tex
- gauge_theory.tex
- group_actions.tex
- guide.tex
- homology.tex
- introduction.tex
- lie_theory.tex
- structures1.tex
- structures2.tex
- topology.tex
.gitignore
bib.bib
custom-numeric-comp2.bbx
geom_intro.tex
preamble.sty
pzccal.sty
README.md

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# Differential Geometry: an introduction by a scenic route

**Download the PDF under [Releases](https://github.com/abogatskiy/Geometry-Intro/releases)**

This is an attempt at an intuitive, bottom-up approach to modern differential geometry, covering deep topics in maximum detail with minimal prerequisites. Too often in math education, fundamental concepts are treated as a mere "means to an end" (such as proving an important theorem), so the original motivation behind the introduction of these concepts gets lost, making them very unintuitive and dry. Therefore, "scenic route" in the subtitle refers to the principle: *no sophisticated construction is introduced until it is motivated enough to feel necessary*[^1].

**Prerequisites**: although a short reference section on point set topology is included, it is not comprehensive. Thus, apart from general calculus, some familiarity with basic topology definitely helps. In certain applications and examples, other subjects such as PDEs, complex analysis, or physics will come in handy.

**Reading guidance**: top-level ``parts'' of the book (Roman numerals) are well separated in themes so that cross-references between them are pretty minimal. Thus, for example, if you are broadly familiar with the topics in parts 1-3, you are probably safe reading from part 4. The text includes a chapter guide with some suggested paths through the book.

##    If you would like to support the continuing work on this book, consider buying me a coffee:  ☕ https://buymeacoffee.com/abogatskiy ☕

Topics covered (the ones with ellipses are under construction)
* Category theory
* Homotopy theory
* Smooth manifolds
* Covering spaces
* Lie groups and algebras
* Fiber bundles, G-structures
* Lie group actions, homogeneous spaces, principal bundles
* Lie's and Cartan's theory of symmetries of ODEs and PDEs
* Cartan's moving frames and equivalence methods
* Cartan's approach to PDEs (Exterior Differential Systems)
* Curves and Surfaces
* Principal connections
* Cartan connections
* Riemannian geometry, Morse theory...
* Complex geometry...
* Symplectic geometry...
* Homological algebra
* Homology and cohomology of topological spaces
* Homology and cohomology of smooth manifolds (de Rham, Hodge, Morse)...
* Sheaf cohomology...
* Structure theory of Lie groups and algebras...
* Classification of fiber bundles, characteristic classes...
* Gauge theory, Dirac bundles...
* Index theorems...

[^1]: I have also referred to this way of thinking as 'autistic', but have since omitted it in the title to prevent misunderstandings. This footnote is meant to serve as an affirmation of neurodivergent modes of thinking that are prevalent but often unrecognized in many areas of science. I believe that neurodivergent people are integral to the scientific pursuit and that their mental well-being demands being recognized and included as such. If nothing else, I hope that this statement will encourage some readers to dive deeper into the beautiful subjects that I tried to assemble in this book in a way that is friendly to those of us who tend to want to understand the essence of common tools before applying them.