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Let $S$ = {$(x,y)\in R^2 | x^2+y^2<1$ } with the first fundamental form $ds^2 = \frac {4}{(1-x^2-y^2)^2}{(dx^2+dy^2)}$


$ds^{2}=Edu^{2}+2Fdudv+Gdv^{2}$

E = G = $\frac {4}{(1-x^2-y^2)^2}$ , F=0

$K={\frac {\det {\mathrm {I\!I}}}{\det {\mathrm {I}}}}={\frac {LN-M^{2}}{EG-F^{2}}},$

How to calculate Second Fundamental form??

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1 Answer 1

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There is no second fundamental form when you have an abstract Riemannian manifold. (But there is nevertheless an intrinsic Gaussian curvature.) Only when your surface is a submanifold of $\Bbb R^3$ (or some other Riemannian manifold) is there a second fundamental form. The second fundamental form measures how the submanifold is twisting inside the bigger manifold.

At any rate, you should look for the formula for the Gaussian curvature just in terms of the first fundamental form. It's rather nice when $F=0$.

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