Editing Second fundamental form (section)

== Hypersurface in a Riemannian manifold ==

In [[Euclidean space]], the second fundamental form is given by

:<math>\mathrm{I\!I}(v,w) = -\langle d\nu(v),w\rangle\nu</math>

where <math>\nu</math> is the [[Gauss map]], and <math>d\nu</math> the [[pushforward (differential)|differential]] of <math>\nu</math> regarded as a [[vector-valued differential form]], and the brackets denote the [[metric tensor]] of Euclidean space.

More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the [[shape operator]] (denoted by {{math|''S''}}) of a hypersurface,

:<math>\mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle = -\langle \nabla_v n,w\rangle=\langle n,\nabla_v w\rangle \,,</math>

where {{math|∇<sub>''v''</sub>''w''}} denotes the [[covariant derivative]] of the ambient manifold and {{math|''n''}} a field of normal vectors on the hypersurface.  (If the [[affine connection]] is [[torsion tensor|torsion-free]], then the second fundamental form is symmetric.)

The sign of the second fundamental form depends on the choice of direction of {{math|''n''}} (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of [[orientability|orientation]] of the surface).

=== Generalization to arbitrary codimension ===

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
:<math>\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,, </math>
where <math>(\nabla_v w)^\bot</math> denotes the orthogonal projection of [[covariant derivative]] <math>\nabla_v w</math> onto the normal bundle.

In [[Euclidean space]], the [[Riemann curvature tensor|curvature tensor]] of a [[submanifold]] can be described by the following formula:
:<math>\langle R(u,v)w,z\rangle =\mathrm I\!\mathrm I(u,z)\mathrm I\!\mathrm I(v,w)-\mathrm I\!\mathrm I(u,w)\mathrm I\!\mathrm I(v,z).</math>
This is called the '''[[Gauss–Codazzi equation|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 {{math|''N''}} is a manifold embedded in a [[Riemannian manifold]] {{math|(''M'',''g'')}} then the curvature tensor {{math|''R<sub>N</sub>''}} of {{math|''N''}} with induced metric can be expressed using the second fundamental form and {{math|''R<sub>M</sub>''}}, the curvature tensor of {{math|''M''}}:
:<math>\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle\,.</math>