So I'm learning surface theory in my differential geometry class. Unfortunetly I've missed a number of classes recently (won't go into details) and the only way to get the notes is through someone, though I don't really know anyone so I figured I'd ask a few questions here to get grounded. These are just a smattering of questions (usually definitions (it's hard to just look up symbols)) so please bear with me and any question you feel you can contribute to I would appreciate greatly!
I... In a problem for this weeks homework the expression "$\nabla E_2 = E_1\omega_{12}$ with $d\omega_{12} = 0$" appears. I want to know...
What is the form "$\omega_{12}$"? I think I saw somewhere something about a 'connection form'? Is this a standard sort of mathematical object?
Also I'd like to know what "$\nabla$" is. If anyone can define it (in a different way than is given, or expand on the given definition) I would really appreciate that.
II... Further in the homework I see "$\nabla$" again (Now with fancy subscript):
It's in the expression "What is $\nabla_X Y$ of two vector fields X and Y". Prior to this $\nabla$ is referred to as the "Levi-Civita Connection"
Again, now in proving that a curve, $\gamma$, satisfied the geodesic equation: "$\nabla_{\gamma'}\gamma'$". So given the previous information, the geodesic equation is just the levi-civita connection wrt to $\gamma'$?
Essentially I need to know what $\nabla_{X}Y$ is. How does $\nabla$ act on the vector field $Y$ (explicitly please) and what does the subscript $X$ denote? Differentiation in a particular direction? I'm assuming $\nabla_{\gamma'}\gamma'$ follows the same rules, just with $\gamma'$.
III... The curvature function (in the context of a Riemannian Manifold) $K_g$ appears a lot as well. If anyone can give me the definition, a link to the definition or anything regarding how I calculate this thing then I would really appreciate that as well.
I recommend for you a quick read in the books: "Riemann Geometry" and "Differential Forms and Applications", both of the author Manfredo do Carmo. There you will find quickly the answers of your questions.