TITLE: Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
QUESTION [14 upvotes]: 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!

REPLY [4 votes]: 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