# shape operator

The *shape operator ^{}* $S$ of a surface $\mathrm{\Sigma}$ in ${\mathbb{R}}^{3}$ is the derivative of the sphere map $N:\mathrm{\Sigma}\to {S}^{2}$ given by $N(p)=$ unit normal

^{}vector field

^{}at $p$. So at each $p$, $S(p)={d}_{p}N$ and it is the linear transformation $S(p):{T}_{p}\mathrm{\Sigma}\to {T}_{N(p)}{S}^{2}$. This is important, because the determinant

^{}defines the Gaussian curvature at $p$ in $\mathrm{\Sigma}$.

Title | shape operator |
---|---|

Canonical name | ShapeOperator |

Date of creation | 2013-03-22 16:04:31 |

Last modified on | 2013-03-22 16:04:31 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 6 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 53A05 |

Related topic | SecondFundamentalForm |