Intuition in some definitions and equations in Riemannian geometry

Question

I'm currently learning the basics of Riemannian geometry. The course notes are based on Do Carmo's book. It introduces many objects without any motivation or intuition. Also on the internet I can't find any decent references explaining the things I need (the best I found is probably the book of Lee: An introduction to curvature). Below I list some of my questions, I hope they will also help other students with the same problem as me.

The following questions are based on the equations of Gauss, Codazzi and Ricci:

$\textbf{1)}$ What is the motivation which Gauss, Codazzi and Ricci had when deriving their formulas? Why do we precisely look at the objects appearing in these equations? At first sight it seems to me that many other relations could be derived. What makes these equations so special?

$\textbf{2)}$ What is the meaning of $[A_{\xi}, A_{\eta}] =A_{\xi}A_{\eta} - A_{\eta}A_{\xi}$? Where $A_{\eta}$ is the shape operator in the direction of the normal vector $\eta$.

$\textbf{3)}$ If we have an immersion $f:M^n \rightarrow \overline{M}^{n+1}$ and denote by $\nabla^{\perp}_X Y$ the normal connection. At page 135 of Do Carmo's book he starts with investigating the properties of $\nabla^{\perp}_X \eta$ where $\eta$ a normal vector field along $M$. For regular surfaces $M$ in $\mathbb{R}^3$, this normal part is zero since the image of the differential of the normal map lies completely in the tangent space of $M$. So what information are we hoping to get when studying this object?

$\textbf{4)}$ What does the normal curvature $$R^{\perp}(X,Y) \xi = \nabla^{\perp}_X \nabla^{\perp}_Y \xi - \nabla^{\perp}_Y \nabla^{\perp}_X \xi-\nabla^{\perp}_{[X,Y]} \xi$$ measure, what is the motivation for introducing it?

Thanks in advance.

Answer 1

Not really an answer, too long for a comment, maybe helpful.

I'm not a differential geometer but I am an admirer of Spivak. In the preface to the first edition of the first volume of his classic A Comprehensive Introduction to Differential Geometry he writes

... no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how any one ever thought of them.

which is why he wrote the book. I think you would enjoy reading it. It's available from many websites. You really ought to buy a copy (if Spivak is still collecting royalties).