How to appreciate Riemannian geometry

Question

I'm currently following an introduction to Riemannian geometry i.e. connections, curvature and isometric immersions (the Gauss, Codazi and Ricci equations).

I find the introduction to Riemannian geometry interesting, but whenever I look at some theorems beyond the introductory topics they seem quite artificial and not intuitive. Also I can't see why they are interesting for us?

There are many examples, one of them is Schur's lemma which goes as follows:

Let $M$ be a Riemannian manifold of dimension $n \geq 3$. Suppose that for every plane $\pi$ in $T_pM$, $K(\pi)$ (the sectional curvature) has the same value $c(p)$. Then $c(p)$ is a constant function.

First the theorem only works in $n\geq 3$, but my main problem is that my intuition lives in surfaces in $\mathbb{R}^3$ where there is only one plane $\pi $ in $T_pM$. Hence what is the intuition behind this lemma? How can one see the beauty of such theorems?

This is however unfortunate since the theory of Riemannian geometry is a popular branch of mathematics which implies many people are interested in it (and probably see the beauty of such theorems and problems). The purpose of my question is to get some intuition or feeling for it so I can appreciate such theorems.

EDIT 1: The very general question, has more concrete subquestions:

1. Why are we interested in the relation between curvature and the shape of manifolds, what is the importance of this?
2. How does one intuitively see which relations (in 1)) one can expect and which not? (for example if the sectional curvatures are $\leq 0$ then for what properties of $M$ can one hope for?)
3. Some theorems hold only in specific higher dimensions, for example Schur's lemma above. How does a mathematician find such theorems and proofs?

EDIT 2: As suggested in a comment, maybe these questions can be answered by giving interesting examples of the uses of Riemannian geometry.

Answer 1

Perhaps this will help, as it is quite intuitive: "Surface in 3D that realizes all pairs of principal curvatures": angel's curl surface:

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Answer 2

In the spirit of answering a small piece of your large question with a visual image (and noting the meta aspect of gradually covering the complicated manifold of Riemannian geometry with local patches of conceptual coordinates), here's the prototypical non-trivial example of parallel transport, illustrating holonomy (parallel transport around a closed loop is not the identity map on the tangent space) and curvature (the holonomy around a geodesic triangle in a surface is the integral of the curvature over the triangle's interior):

Answer 3

The first book you should look at is Vladimir Arnold's Mathematical Methods of Classical Mechanics where he has a nice introductory discussion of differential geometry and curvature.

The lemma you cited does not have far-reaching consequences and you shouldn't be focusing on it. One direction of research that is quite popular is the relation between curvature and topology. It became clear relatively recently (in the 1980s) that positive sectional curvature imposes extremely stringent conditions on the manifold; e.g., one gets a universal upper bound on the sum of all Betti numbers of the manifold by a result of Gromov. In negative curvature, on the contrary, there is a great wealth of examples, related also to the popular field of Cannon-Gromov-hyperbolic groups. In general, to get motivated I would suggest looking up work by Gromov. You may not follow all the details (if the details are there :-) but you are likely to be inspired.