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

Abstract

A simple formula is derived for the Ricci scalar curvature of any smooth level set ${\psi(x_0,x_1,...,x_n)=C}$ embedded in the Euclidean space $ \mathbb R^{n+1}$, in terms of the gradient $ \nabla\psi$ and the Laplacian $ \Delta\psi$. Some applications are given to the geometry of low-dimensional $p$-harmonic functions and high-dimensional harmonic functions.