Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology 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

*

*Riemannian manifold has a good cover.


*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.


*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!
 A: 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”.
A: 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

A: 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.
A: 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:

*

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


*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.


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


*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.
A: 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).
A: 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.
