I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. I couldn't wish for a better guide to the details.

But I feel like I could use a one-page overview of how these unfamiliar objects (manifolds, tensors, forms, Lie groups, frames, fibre bundles etc) fit together into a system; something to help me see the forest before I start prodding specific trees. I understand the basic definitions but lack a picture that shows how they fit together.

To re-iterate, I'm looking for an extremely high-level map only. Something I can print out and pin to my wall. Is such a thing possible? If it is, I feel sure someone must already have done it...

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    $\begingroup$ this isn't an answer, of course, but I often find myself thinking: wouldn't it be great if there were a book, or a website, where you could find a comprehensive set of high-level maps for all the major areas of modern mathematics... (I don't think the Princeton Encyclopedia counts.) many posts here on stackexchange would count, as would several blogposts around the internet, but they should be curated and published somewhere easier to find. $\endgroup$ – symplectomorphic May 17 '14 at 10:50
  • $\begingroup$ @symplectomorphic -- Yes, I agree entirely. Maybe one day I'll understand enough to make some... it'd be a lot of fun. $\endgroup$ – helveticat May 17 '14 at 11:34
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    $\begingroup$ FWIW, I learned a lot about high-level maps for differential geometry and topology from John Baez's websites. consider exploring those if you haven't. $\endgroup$ – symplectomorphic May 17 '14 at 11:38
  • $\begingroup$ Interesting, thanks -- I was just browsing there and saw he recommended Dieudonne's Panorama of Pure Mathematics, which I'd forgotten existed. Might be worth a quick spin through the relevant bits. $\endgroup$ – helveticat May 17 '14 at 12:09
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    $\begingroup$ This isn't an answer other, but I have to recommend two books by Marcel Berger: A Panoramic View of Riemannian Geometry and Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry. While the first book is not exclusively about differential geometry, it is incredibly lucid and informative, especially as it pertains to providing insight into the types of questions that drive the field. $\endgroup$ – THW May 17 '14 at 17:35

This is what I came up with after about ten minutes of doodling. As others have said, this will inevitably be missing things, but this is probably good to get you started. Note that my bias is mine, and does not necessarily reflect everyone else's views.

Also, note that "tangent spaces" was cut off at the bottom. It links to vector fields, distributions, frames, and Riemannian geometry.

Hand-drawn Overview of "The Basics"

  • $\begingroup$ I want back to look at this lovely picture again today and realised I'd neither accepted your answer nor thanked you. So, very belatedly: apologies for my neglect, and thank you! $\endgroup$ – helveticat Jan 22 '18 at 19:00

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