I am looking to calculate the Riemann curvature on an orthonormal frame for the tangent bundle of the surface parametrized by the equation $\phi: \mathbb{R}^2\to \mathbb{R}^3$ given by $$ (x,y) \mapsto (x,y,f(x,y))$$ for some continuous $f:\mathbb{R}^2 \to \mathbb{R}$. I would like to do this by calculating the Christoffel symbols with respect to the orthogonal tangent vectors $$ \begin{align} \frac{\partial}{\partial \varphi_1} &= -f_y\frac{\partial}{\partial x} + f_x\frac{\partial}{\partial y} \\ \frac{\partial}{\partial \varphi_2} &= f_x\frac{\partial}{\partial x} + f_y\frac{\partial}{\partial y} + (f_x^2 + f_y^2)\frac{\partial}{\partial z} \end{align} $$ (following the approach of Orthonormal basis for a tangent plane) where the metric is the restriction of the Euclidean metric from $\mathbb{R}^3$. I would then calculate $\nabla_\frac{\partial}{\partial \varphi_i}\frac{\partial}{\partial \varphi_j}$ for all $i,j$ and hence the curvature on the normalised frame. However, I am unclear on a few points:
- If $z = f(x,y)$, then should I calculate $\frac{\partial}{\partial z} = f_x \frac{\partial}{\partial x} + f_y \frac{\partial}{\partial y}$ (where $f_x = \frac{\partial f}{\partial x}$, etc.)? Doesn't this mess up orthogonality?
- Should I somehow try to rewrite $x,y$ in terms of $\varphi_1, \varphi_2$ and then do the Christoffel symbol calculation? My only previous example was calculating the Riemann curvature for a sphere where the parametrization was in terms of spherical polar coordinates, which made the calculus cleaner.
- There is an expression for Christoffel symbols in terms of $\phi_x, \phi_y, \phi_{xx}$, etc. as per Christoffel Symbols on a Surface. However, here $\phi_x$ and $\phi_y$ are not orthogonal, so would it still be possible for me to calculate Christoffel symbols using that method and then apply those to the orthogonal frame I wish to use?
EDIT MADE: Corrected the choice of tangent vectors in the orthonormal frame, as the original frame was not tangent.
I am currently struggling with surfaces as well at the moment, so I'm no expert, take this answer with a grain of salt. As you're correclty stating $\phi_x , \phi_y$ are NOT an orthonormal frame for $T_p S$, though I think that it is much easier to express Christoffel symbols in terms of those basis vectors. Using the system $$\phi_{xx} = \Gamma_{11}^1 \phi_x + \Gamma_{11}^2 \phi_y + L^1 N$$ $$\phi_{xy} = \Gamma_{12}^1 \phi_x + \Gamma_{12}^2 \phi_y + L^2 N$$ $$\phi_{yy} = \Gamma_{22}^1 \phi_x + \Gamma_{22}^2 \phi_y + L^3 N$$ and multiplying each equation scalarly by $\phi_x$ and $\phi_y$ the normal vector $N$ always cancels out (being orthgonal to $T_p S$), and you're left with a linear system of six equations from which you can derive the six Christoffel Symbols.