Mixing
Second fundamental form of a hypersurface with a defining function
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The matrix of the second fundamental form of a parametrized hypersurface is given by $$h_{ij}=-langle N,r_{ij}rangle,$$ where $N$ is the unit normal vector and $r$ is the parametrization of the hypersruface (say $M$ of $mathbb{R}^N$ ). If the same hypersurface has a defining function $rho:mathbb{R}^Ntomathbb{R}$ , with $left| nabla rhoright|=1$ along $M$ , then the unit normal is given by $$N = nablarho.$$ The shape operator is given by $$S_p = -dN_p = -Hessian_p(rho).$$ Is the matix of the second fundamental form of $M$ at $p$ is the same as the hessian of $rho$ ? How does the quadratic form of the second fundamentatl form look like?
differential-geometry
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