[Riemannian Geometry] A Guide to Study

One may be curious why Ricci's calculus helps us to understand about topology of manifolds (a very famous example is the proof of Poincare's conjecture). This topic is introduced in the book of Do Carmo on Riemannian Geometry, where he discuss how one can use information of curvatures and geodesics to deduce facts about topology. It is really interesting !!! One can try to begin reading Riemannian geometry in Do Carmo, a very beautifully written book on the subject.

My first advice is, if you don't have background on differential geometry of surfaces, DON'T try to read this. Otherwise, you first impression of Riemannian geometry is that you fear it, and it is a really bad feeling. For example, one can see some kinds of formula of the following form in the book

$$R^s_{ijk}=\sum_l\Gamma^l_{ik}\Gamma^s_{jl}-\sum_l\Gamma^l_{jk}\Gamma^s_{jl}+\frac{\partial}{\partial x_j}\Gamma^s_{jk}-\frac{\partial}{\partial x_i}\Gamma^s_{jk}$$

It is a formula in Ricci's calculus to compute curvatures, looks very complicated (and it actually is). Now, one should take Do Carmo after he understands well about differential geometry of surfaces. I highly recommend the book of K. Tapp "Differential Geometry of Curves and Surfaces", especially his Chapter V about geodesics. He treats surfaces with modern language of Riemannian geometry, for example: covariant derivative, parallel transport, exponential mapping, and Jacobi fields.

All of these notions are generalized later on Do Carmo, with the help of the new fundamental notions: curvatures (Ricci's calculus). One does not need to know about vector bundles to learn Riemannian Geometry, all you just know is that how $T_pM, TM$ is defined/constructed and what is a vector field, and how it acts as a derivation. These things are very basic, and can be found in any book about differential geometry.

After equipping sufficient knowledge on surfaces, and differential geometry as I mentioned above, you can now begin reading Do Carmo. His book should be now readable and geometric pictures may appear on your mind while reading (though like algebraic geometry/complex geometry textbooks, there are very little figures in these kinds of geometry-all you need are specific cases, and your imagination).

You may be amazed that why sheaves and cohomology do not appear in the book. The answer is one often use sheaves and cohomology when problems are linear, or its solution can be found easily locally. But when we deal with Riemannian geometry, local problems are often appear as differential equations, and they are, for sure, very hard to solve completely.

Here are some words for someone who likes geometry. Have fun !!!