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I'm now attending a reading seminar on the algebraic topology.

The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).

In those books, theorems on the Riemannian manifolds are frequently just mentioned and used.

To mention some examples

  1. Riemannian manifold has a good cover.

  2. Exponential Map is used to find a tubular neighborhood for a pair of manifolds. (where one is a submanifold of the other ) and its properties are used in computations on the dual cohomology class and the diagonal cohomology class.

  3. Argument in a proof which states that we can reduce the general case to a local open submanifold with the Euclidean standard metrics.

and maybe more.

When I browse books on differential geometry or Riemannian manifolds, I get the feeling that I cannot avoid studying the standard materials like the connection, tensors...

But I have no time to study all that materials.

Is there some shortcut to understand those materials (at least for good manifolds) without studying all the details of these differential materials? (Maybe is there some axiomatic approach?)

Any suggestions on the references are welcome.

Thank you very much!

David Lehavi
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gualterio
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    The first chapter of Cheeger-Ebin „Comparison Theorems in Riemannian Geometry“ might give what you want, i.e., a quick introduction to the basic properties of Riemannian manifolds (though I‘m not sure that existence of good covers is treated there). – ThiKu Jan 06 '21 at 09:45
  • @ThiKu Thank you very much for your suggestions. It seems that the standard concepts are needed afterall, but the exposition seems to be succint! – gualterio Jan 06 '21 at 10:11
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    If you have a time machine you can read the book and then go back to the present day. You can also research neural link and create a device that would load that knowledge directly into your brain (olfactory conditioning seems like a promising field, although maybe just for emotional response). Then just go back to now, and load the knowledge directly into your brain and go back to the future. (Yes, this will create a closed time loop, but where's your sense of adventure?) – Asaf Karagila Jan 06 '21 at 18:01
  • The existence of good covers has been extensively discussed on MathOverflow, see, for example, https://mathoverflow.net/questions/102161/proving-the-existence-of-good-covers and https://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbbr/212595#212595. The results of these discussions are assembled on the nLab in the article https://ncatlab.org/nlab/show/good+open+cover, see the references to Guillemin–Haine, Demailly, and Gonnord–Tosel. – Dmitri Pavlov Jan 06 '21 at 23:01
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    It seems to me that if you can identify the specific theorems whose proofs rely on a Riemannian metric, then you can for now accept them to be true and learn the proofs later. These are all technical statements that basically say that the local structure of a manifold is very simple. I consider them to be the nonlinear equivalent of basic facts about vector spaces such as "any finite dimensional vector space has a basis and therefore is isomorphic to $\mathbb{R}^n$. – Deane Yang Jan 07 '21 at 01:11
  • @DeaneYang Thank you for the comment! Could you please explain to me the intuitive picture of the simple local structure? I can just guess that it is a generalization of the classification of 2-dimensional surfaces by the curvature. – gualterio Jan 08 '21 at 16:58
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    I just posted a more detailed answer below. – Deane Yang Jan 08 '21 at 23:56
  • As for Milnor-Stasheff, the only serious use of Riemannian geometry I found is in Appendix C, where the Chern-Weil description of characteristic classes in terms of curvature is presented. It appears that they make an effort to provide details of anything they need. It is true, however, that this section is much easier to read, if you are already familiar and comfortable with the concepts of the Levi-Civita connection and Riemann curvature, as well a the machinery of differential forms. – Deane Yang Jan 09 '21 at 00:06
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    On other hand, little is needed beyond the definitions of the connection curvature and, mostly importantly, the symmetries of the curvature tensor. Beyond that, there are intricate tensor calculations you need to work through carefully. I don't think studying a textbook on Riemanniabn geometry will help much with this. – Deane Yang Jan 09 '21 at 00:09
  • @Deane Yang Thank you very much! – gualterio Jan 09 '21 at 16:48
  • Milnor deserves the credit! Deane, `` there are intricate tensor calculations you need to work through carefully.'' such as?? and where tensor means a thing with indices? – Jim Stasheff Jan 12 '21 at 19:46
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    The good cover stuff is dealt with, not using Riemannian geometry, in my lectures notes: https://cora.ucc.ie/items/274ec834-ca2e-4885-922f-e353d539ef18. – Ben McKay Jun 18 '23 at 12:46

6 Answers6

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Bröcker and Jänich's "Introduction to Differential Topology" is admirably short and nicely written, and you may find that it contains everything that you need.

Neil Strickland
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The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below).

In general, if your main goal is to study (algebraic) topology of manifolds, you probably don't need to know much about metrics and connections and that sort of thing that typical differential geometry books spend a lot of time on. It sounds like what you really need is some material on differential topology. I don't know how much they'll cover of the specific things you're looking for, but here are a few suggestions to check out:

  1. Topology and Geometry by Glen Bredon. This might be a particularly good book for you as it really combines the two topics pretty well.

  2. Michael Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1. Despite the title, the first volume is more about differential topology than geometry. Also this is what Bott and Tu cite for their key fact needed for the existence of good covers.

  3. Differential Topology by Guillemin and Pollack - this is a very readable introduction.

  4. Introduction to Smooth Manifolds by John M. Lee - this is oriented a bit more toward geometry but you can find a lot in it.

I'd recommend skimming through these (and others) to find one that suits you and has the kinds of things you're looking for.

Greg Friedman
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There's a short book by Milnor: "Topology from a differentiable viewpoint" (here is one link https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf where you can find it).

David Lehavi
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    I don‘t think that properties of Riemannian manifolds are treated in that book at all. – ThiKu Jan 06 '21 at 09:42
  • @David Lehavi Thank you for the suggestion. But as ThiKu has commented the material of the book seems to be rather on the general differential geometry. – gualterio Jan 06 '21 at 10:13
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    @ThiKu Indeed they aren't. However, it includes a lot of what you do need for Bott & Tu: e.g. the homeomorphism between a small neighborhood of a point and a 0-neighborhood in the tangent space; tubular neighborhoods, etc. For the purposes of Bott & Tu, I think it's enough - you don't really need this morphism to be the exponential map. – David Lehavi Jan 06 '21 at 12:03
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    Great suggestion. One might underestimate this book because it's short and relatively "low-tech." But once you've read it, a lot of differential topology seems obvious rather than mysterious. It's a masterpiece of mathematical exposition. – Martin M. W. Jan 07 '21 at 15:46
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    Typical Milnor! – Jim Stasheff Jan 12 '21 at 19:49
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To answer the first question: there are two completely elementary proofs of the existence of differentiable good open covers: the first one is Lemma IV.6.9 in Demailly's “Complex Analytic and Differential Geometry” and the second one is Theorem 5.3.2 and Appendix C in Guillemin and Haine's “Differential Forms”. These proofs rely on a fact that star-shaped open subsets of R^n are diffeomorphic to R^n, an an elementary proof of which can be found here: What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$.

To answer the second question: for an elementary proof of the existence of tubular neighborhoods, see Theorem 4.5.2 in Hirsch's “Differential Topology”.

Dmitri Pavlov
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I just searched "Riemann" in Bott-Tu and scanned through most occurrences. Proving the existence of a good cover requires the fact that each point in a Riemannian manifold has a geodesically convex neighborhood. You can probably find the proof of this in most textbooks on Riemannian geometry. You can also safely just assume the existence of a good cover without learning the proof.

After that, all uses of Riemannian manifolds appear to rely on only the definition of a Riemannian metric and nothing more. Neither the Levi-Civita connection nor the curvature tensor are ever needed. Also, the book provides full details of any proof that uses a Riemannian metric.

Here are the uses of either a Riemannian metric or its equivalent on a vector bundle that I found:

  • Two consist of the reduction of the tangent bundle into a sphere bundle and the reduction of the frame bundle into an orthonormal frame bundle.
  • The construction of the global angular form
  • The definition of a "radial function"
  • The definition of the gradient of a function
Deane Yang
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My favourite is Kosinski's "Differential manifolds". It is short, has many details often overlooked and it costs just 10$.

The first part (pg 1-75) covers transversality and tubular neighbourhood (the things you mentioned). In the second part there is rigorous treatment of handle theory while the third part covers the h-cobordism theorem and some surgery theory.

Roberto Ladu
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  • Kosinski's book constructs tubular neighborhoods using exponential maps on Riemannian manifolds, which the original poster indicated as an undesirable method of proof. – Dmitri Pavlov Jan 11 '21 at 21:11