Application of Gauss

In traditional differential Geometry, the application of Gauss is a natural application differentiable on a surface of $R^3$, with values in the sphere unit $S^2$, and whose differential gives access the Second fundamental form. It holds its name of the German mathematician Carl Friedrich Gauss.

Application of Gauss

That is to say $\backslash \; Sigma$ a directed Surface of class $C^\; \{k+1\}$ of $R^3$.

For $P$ a point of $\backslash \; Sigma$, there exists a single unit normal vector $\backslash \; Gamma\; (P)$. The application of Gauss is the application of class $C^k$:

$\backslash \; Gamma:\; \backslash \; Sigma\; \backslash \; rightarrow\; S^2\; \backslash ,$

One has a natural identification:

$T\_P\; \backslash \; Sigma=T\_\; \{\backslash \; Gamma\; (P)\}S^2\; \backslash ,$

Endomorphism of Weingarten

The Differential of the application of Gauss, seen like linear Operator of $T\_P\; \backslash \; Sigma$, is a symmetrical operator (called endomorphism of Weingarten ) whose quadratic Forme associated is the Second form fundamental $II\_P$ of $\backslash \; Sigma$ in P .

In a more precise way, for any tangent vector $w\; \backslash \; in\; T\_P\; \backslash \; Sigma$, one a:

$II\_P\; (W)\; =\; \backslash \; langle\; D\; \backslash \; Gamma\; (P)\; W|W\; \backslash \; rangle\; \backslash ,$

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