Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian geometry, ie. notions of curvature, connections, riemann tensor.

Is there an algebraic geometry text that has a more thorough review of these notions over $\mathbb{R}$, that reviews the subject comprehensively, but briefly in the first or so chapters? I've taken the course before, and I don't want to have to refer to a bigger text devoted to differential/riemannian geometry and I'd rather just read a brief review of the subject and the main/essential points.

Thanks!

Solution 1:

The best complete, concise, clear and rapid overview of the essentials of Riemannian geometry that I know is found in the little book Morse Theory, by John Milnor. As far as delivering a more detailed review of this book: well, I've already said Milnor; need I say more?

Note Added Wednesday 8 July 2015 9:33 PM PST: Despite my previous invocation of the honored name Milnor as a more or less complete justification of the utility and quality of his Morse Theory as a most excellent source for quickly grasping the basic facts of Riemannian geometry, I felt it might be helpful to add a little bit about the contents of this work, if for no other reason than to fully answer our OP's question, and thus if possible save her or him some time in finding the right place to learn what he/she needs to know. The book, as the title indicates, is directed towards Marston Morse's theory of critical points, an essential tool of differential topologists; but it focuses in the large on the Morse theory of geodesics, applying his critical point theory to the relationship between convergence/divergence of geodesics and manifold topology. In this book, as elsewhere, the critical-point analysis is used to discover facts about the the topology of path spaces, and ultimately to prove the Bott periodicity theorems to reveal aspects of the structure of the higher homotopy groups of the Lie groups $O(n)$ and $U(n)$ etc. This is done via the study of conjugate points, i.e., points along a given geodesic where Jacobi fields, that is, those vector fields $W(t)$ along the geodesic $\gamma(t)$ satisfying

$\nabla_{\dot \gamma} \nabla_{\dot \gamma} W + R(\dot \gamma, W)\dot \gamma = 0, \tag{1}$

where $R$ denotes the Riemann tensor, vanish, having been initialized with $W(0) = 0$, $\nabla_{\dot \gamma}W$ arbitrary.

Now I know very little about algebraic geometry; indeed, I am learning someting of it, slowly and surely, mainly by reading posts here and on Math Overflow. But I suspect that the main utility of Riemann's geometry to the algebraic kind lies in the theory of characteristic classes, which quantify certain important properties of tangent and other vector bundles; here the Riemann tensor field plays an essential role, e.g. the Euler cohomology class, whose evaluation on the top homology class $[M]$ of the manifold $M$ under study, yields the Euler-Poincare characteristic number, may be expressed locally as a polynomial in the components of $R$; the Gauss-Bonnet theorem is the perhaps the most elementary example of this result. So perhaps Milnor's book goes in a slightly different direction than would best suit the needs of an algebraic geometer; but in terms of learning the basics, it is, in my humble opinion, hard to beat. And after reading Morse Theory, one can always tackle Milnor's Characteristic Classes, which may indeed be more suited to the needs of those pursuing algebraic geometry. End of Note.