Second fundamental form

of a surface

The quadratic form in the differentials of the coordinates on the surface which characterizes the local structure of the surface in a neighbourhood of an ordinary point. Let the surface be given by the equation

$$ \mathbf r = \mathbf r ( u, v), $$

where $ u $ and $ v $ are internal coordinates on the surface; let

$$ d \mathbf r = \mathbf r _ {u} du + \mathbf r _ {v} dv $$

be the differential of the position vector $ \mathbf r $ along a chosen direction $ d u / d v $ of displacement from a point $ M $ to a point $ M ^ \prime $( see Fig.). Let

$$ \mathbf n = \ \frac{\epsilon [ \mathbf r _ {u} , \mathbf r _ {v} ] }{| [ \mathbf r _ {u} , \mathbf r _ {v} ] | } $$

be the unit normal vector to the surface at the point $ M $( here $ \epsilon = + 1 $ if the vector triplet $ \{ \mathbf r _ {u} , \mathbf r _ {v} , \mathbf n \} $ has right orientation, and $ \epsilon = - 1 $ in the opposite case). The double principal linear part $ 2 \delta $ of the deviation $ P M ^ \prime $ of the point $ M\prime $ on the surface from the tangent plane at the point $ M $ is

$$ \textrm{ II } = 2 \delta = (- d \mathbf r , d \mathbf n ) = $$

$$ = \ ( \mathbf r _ {uu} , \mathbf n ) du ^ {2} + 2 ( \mathbf r _ {uv} ,\ \mathbf n ) du dv + ( \mathbf r _ {vv} , \mathbf n ) dv ^ {2} ; $$

it is known as the second fundamental form of the surface.

Figure: s083700a

The coefficients of the second fundamental form are usually denoted by

$$ L = ( \mathbf r _ {uu} , \mathbf n ),\ \ M = ( \mathbf r _ {uv} , \mathbf n ),\ \ N = ( \mathbf r _ {vv} , \mathbf n ) $$

or, in tensor notation,

$$ (- d \mathbf r , d \mathbf n ) = \ b _ {11} du ^ {2} + 2b _ {12} du dv + b _ {22} dv ^ {2} . $$

The tensor $ b _ {ij} $ is called the second fundamental tensor of the surface.

See Fundamental forms of a surface for the connection between the second fundamental form and other surface forms.

References

[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) MR0350630 Zbl 0264.53001