I'm some new here, sorry if my question is not correct. I need to calculate the sectional curvature of a Riemannian manifold. I found this formula in several books.
$$K(\sigma,p)=\frac{R(X,Y,Y,X)}{g(X,X)g(Y,Y)-g(X,Y)}$$
My problem is that none of them do the detailed accounts, and it frustrates me because most that I have been able to read have a different notation. I am quite confused, I would like to know if anyone knows any book where I can find some text that makes a summary of the formulas (from defining the metric, Christoffel symbols, Riemannian tensor, etc.) and applies them in an example to understand. And if someone would be so kind to do it here, I would be very grateful, it does not matter if it is a simple example such as the sphere or hyperbolic space or a surface of revolution I don't know... I just want to see the technique to be able to understand and apply it.