So we want to...
i.) Calculate the Levi-Civita Connection of two vector fields X,Y: $$\nabla_{X}Y$$ for the standard euclidean metric in n-dimensions: $$ds^{2} = dx_{1}^{2} + ... + dx_{n}^{2}$$
ii.) Calculate the curvature function of this metric for n = 2.
So for...
i.) I have for the covariant derivative of two vector fields, $V = v^{i}e_{i}$ and $U = u^{j}e_{j}$ that $\nabla_{V}U$ = $v^{i}u^{j}\Gamma^{k}_{ij}e_{k} + v^{i}\frac{du^{j}}{dx^{i}}e_{j}$
- However looking online I found an example where it looked like all he was doing was calculating the christoffel symbols. I was going to calculate $\Gamma^{i}_{ii}, \Gamma^{i}_{ij} = \Gamma^{i}_{ji}, \Gamma^{i}_{jj}, \Gamma^{j}_{ii}, \Gamma^{j}_{ij} = \Gamma^{j}_{ji}$ and $\Gamma^{j}_{jj}$ and since for the euclidean metric only the diagonal (i.e., i=j) elements are 1 with all else 0 and since the symbols all involve derivatives of metric elements, thus making all the symbols equal to zero right?
So should I just calculate those Christoffel Symbols for the Levi-Civita Connection (Bothered because doesn't involve Vector Fields at all)? Or should I use that equation for $\nabla_{V}U$ (in which case the term with christoffel symbols vanish leaving only second term and then I don't know how to make a meaningful answer out of components of arbitrary vector fields)?
ii.) For this I just need to know what the curvature function is. I know theres a normal curvature which looks to be defined as: $$ K_{N} = <c''(0), N(p)>$$ for a curve c with c(0) = p but I've seen it as $$K_{N} = II(c'(0), c'(0)) where II is 2nd fundamental form.
I'm not exactly sure how to calculate something with the Second fundamental form if anyone can elucidate this I would appreciate that.
Basically I just need to know what formula I should be using to calculate the curvature in n = 2.
Yes, since all the $g_{ij}$ are constant, all the Christoffel symbols will be $0$. Your formula for normal curvature is not what is going on here (for starters, we have an abstract Riemannian metric, not a surface in $\Bbb R^3$; next, we want Gaussian curvature, not normal curvature, even in the setting of a surface in $\Bbb R^3$). Do you have a formula for the curvature tensor in terms of things like $\nabla_X\nabla_Y Z$ ... or, perhaps, in terms of the Christoffel symbols and their derivatives?