# A fast introduction to Psuedo-Riammanian Manifolds and Riemannian Manifolds

I apologize in advance for the long length of this question

Okay so let me explain my background. I know quite a bit of General Topology, including Topological Manifolds. I know some basic stuff about Algebraic Topology, and Differential Topology (about half of Milnor's Topology from the Differentiable Viewpoint and the half of the first chapter of Guillemin and Pollack)

My strength is Topology, but I'm currently trying to to read up on a paper that deals with a topological construction on smooth manifolds. I can understand the topological aspects just fine, however there are a number of examples given of this construction on Pseudo-Riemannian Manifolds, of which I've never dealt with and I need to understand these examples of Pseudo-Riemannian examples to be able to fully understand the paper (which is geared towards the application of this topological construction to Pseudo-Riemannian Manifolds).

Now I need to learn this material fast, as the research group I'm working with hopes to publish a paper sometime in June next year. I'm giving myself around 1 month to learn the pre-requisite Riemannian and Pseudo-Riemannian Geometry

I'm currently reading through Lee's Introduction to Smooth Manifolds, and I'll soon be fairly comfortable with the first 3 chapters. What's the fastest way to learn enough Riemannian and Pseudo-Riemannian Geometry to understand the examples given below provided that I have no knowledge of

• tensors / tensor fields
• differential forms
• vector bundles and tensor bundles

But I do have a solid knowledge of metric spaces, metrics (not riemannian metrics, just metrics on metric spaces), and linear algebra and basic group, ring and module theory. For example I find Keith Conrad's notes on tensor products of modules quite readable (http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf)

What chapters (and in what order) should I look at in Lee's Introduction to Smooth Manifolds to gain a solid grasp of Pseudo-Riemannian Geometry and Riemannian Geometry?

If there is another book which is more streamlined to Riemannian Geometry and Pseudo-Riemannian Geometry, and which would allow me to learn the material faster, please let me know. The two disadvantages to this is that I'd have to get used to another authors notation (which is really not an easy process especially in Differential Geometry), and I wouldn't be using Lee's book which I plan to use for DIfferential Geometry.

Below is the level of Riemannian Geometry that I need to learn, it's an example taken from the paper.

• For Riemannian geometry: do Carmo's "Riemannian Geometry". For semi-Riemannian geometry- O'Niel's book amazon.com/…. – Moishe Kohan Dec 6 '17 at 14:50
• From memory, Lee's book that you mention doesn't contain much Riemannian geometry. Since you're concerned about using the same conventions, you could perhaps use his Riemannian Manifolds.... Really, though, you need to get used to different notations and conventions if you plan to be doing much differential geometry. – Anthony Carapetis Dec 6 '17 at 23:17