Suppose I have a 2D surface $S=(x,y,f(x,y))$ embedded in a Riemannian manifold $(M,g)=(\mathbb{R}^3,g)$, i.e. in a curved 3-space.
Then the induced metric tensor $h$ on $f$ is $ h_{ij} = g_{ab}\, \partial_i S^a \partial_j S^b $ in local coordinates.
Now I want the surface normals. Normally one takes the cross-product of the tangent vectors. What is the Riemannian analogue here?