Let \(f:R^ d\to R\) be a \(C^ 2\) function and let \(V=f^{-1}(c)\) be a level surface on which grad f(x) is never zero and orient V with the field n(\(\cdot)\) of normal vectors. Let H(x) be the mean curvature at x. We prove the following: 1. A process X in \(R^ d\) with \(f(X_ 0)=c\) and \(dX=dB n(X)+2^{-1}(d- 1)H(X)n(X)dt\) is a Brownian motion on the level surface \(V=f^{-1}(c).\) 2. A process X in \(R^ d\) with \(f(X_ 0)=c\) is a Brownian motion on the level surface \(V=f^{-1}(c)\) if and only if X is a continuous semimartingale such that (i) \(dX-(d-1)H(X)n(X)dt=dM\), where M is a local martingale, (ii) \(d=\{\delta_{ij}-n^ i(X)n^ j(X)\}dt\).