A Simple Formula for Scalar Curvature of Level Sets in Euclidean Spaces

A simple formula is derived for the Ricci scalar curvature of any smooth level set $\psi (x_0,x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=C$ embedded in the Euclidean space $ \mathbb R^{n+1}$,intermsofthegradient$ \nabla\psi$andtheLaplacian$ \Delta\psi$\mathrm{.}Someapplicationsaregiventothegeometryoflow-dimensional$p$-harmonic functions and high-dimensional harmonic functions.