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!