I have studied differential geometry, and am looking for basic introductory texts on Riemannian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, connections and transport belong more firmly in Riemmanian geometry.
I am aware of earlier questions that ask for basic texts on differential geometry (or topology). However, these questions address mainly differential geometry. I'm more interested in Riemannian geometry here.
Best Answer
Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it is a genuine 'introduction' (as opposed to some textbooks which just seem to almost randomly put the word on the cover).
However, right from the first line: "If you've just completed an introductory course on differential geometry, you might be wondering where the geometry went", I was hooked. It introduces geodesics and curvature beautifully and is very readable.
I think the first chapter might be available on the author's website.