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