# Computing the inverse Riemannian metric of graph of a function in graph coordinates

For an open set $$U\subseteq \mathbb{R}^n$$, given a smooth function $$f:U\to \mathbb{R}$$, its graph $$G_f = \{(x,f(x)) ~|~ x\in U )\} \subseteq \mathbb{R}^{n+1}$$ is a hypersurface (codimension-1 submanifold) in $$\mathbb{R}^{n+1}$$. The coordinate chart on $$G_f$$ corresponding to the parametrization $$X:U\to G_f$$ given by $$X(u) = (u,f(u))$$ is called its graph coordinates. The coordinate vectors here are $$X_*\left(\partial_i\right) = \frac{\partial}{\partial x_i} + \partial_i f ~\frac{\partial }{\partial x_{n+1}}$$ and so the metric coordinates are $$g_{ij} = \delta_{ij} + (\partial_i f)( \partial_j f)$$

The claim is that its inverse is $$g^{ij} = \delta^{ij} - \frac{ (\partial_i f) (\partial_j f) }{ 1+\|\nabla f\|^2}$$ where $$\| \cdot \|$$ denotes the Euclidean norm.

I have checked this for $$n=1$$ but I don't know how to proceed from there.

Maybe use the so-called Sherman–Morrison formula

https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula

with $$A={\rm Id}_{n\times n}$$ (the identity matrix), and where $$u=v=\nabla f$$.

• This works. Thank you! Commented May 1 at 6:45