SECOND FUNDAMENTAL FORM

In differential geometry, the 'second fundamental form' is a quadratic form, usually denoted by II, on the tangent space of a hypersurface in a Riemannian manifold, such as a surface in three dimensional Euclidean space. It is an equivalent way to describe the shape operator (denoted by ''S'') of a hypersurface,
:$mathrm\; I!mathrm\; I(v,w)=langle\; S(v),w$
angle= -langle
abla_v n,w
angle=langle n,
abla_v w
angle,
where $$
abla_v w denotes the covariant derivative and ''n'' a field of normal vectors on the hypersurface.
The sign of the second fundamental form depends on the choice of direction of ''n'' (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).
The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by
:$mathrm\{I\}!mathrm\{I\}(v,w)=($
abla_v w)^ot,
where $($
abla_v w)^ot denotes the orthogonal projection of covariant derivative $$
abla_v w onto the normal bundle.
In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:
:$langle\; R(u,v)w,z$
angle =langle mathrm I!mathrm I(u,z),mathrm I!mathrm I(v,w)
angle-langle mathrm I!mathrm I(u,w),mathrm I!mathrm I(v,z)
angle.
This is called the 'Gauss equation', as it may be viewed as a generalization of Gauss's Theorema Egregium.
For general Riemannian manifolds one has to add the curvature of ambient space; if ''N'' is a manifold embedded in a Riemannian manifold (''M,g'') then the curvature tensor $R\_N$ of ''N'' with induced metric can be expressed
using the second fundamental form and $R\_M$, the curvature tensor of ''M'':
:$langle\; R\_N(u,v)w,z$
angle = langle R_M(u,v)w,z
angle+langle mathrm I!mathrm I(u,z),mathrm I!mathrm I(v,w)
angle-langle mathrm I!mathrm I(u,w),mathrm I!mathrm I(v,z)
angle.

Contents

Further Notation

See also

Further Notation

The second fundamental form is often written in the modern notation of the metric tensor.
:$II\; =\; L\; du^2+2Mdu\; dv+Ov^2$.
The coefficients may then be written as $b\_\{ij\}$:
:$left(b\_\{ij\}$
ight) = egin{pmatrix}b_{11} & b_{12} \b_{21} & b_{22}end{pmatrix} =egin{pmatrix}L & M \M & Oend{pmatrix}
Where
The coefficients of the first fundamental form may be found by taking the dot product of the second partial derivatives with the unit normal.
:$L\; =\; X\_\{uu\}\; cdot\; N$
:$M\; =\; X\_\{uv\}\; cdot\; N$
:$O\; =\; X\_\{vv\}\; cdot\; N$
Normal:
:
: where $I$ is the First fundamental form defines the normal curvature of the curve
The second fundamental form describes how a surface varies from its tangent plane.

See also

★ First fundamental form

★ Gaussian curvature