Third fundamental form: Difference between revisions

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In differential geometry, the third fundamental form is a surface metric denoted by $\mathrm {I\!I\!I}$ . Unlike the second fundamental form, it is independent of the surface normal.

Definition

Let $S$ be the shape operator and $M$ be a smooth surface. Also, let $u p$ and $v p$ be elements of the tangent space $T p (M)$ . The third fundamental form is then given by

\mathrm {I\!I\!I} (\mathbf {u} _{p},\mathbf {v} _{p})=S(\mathbf {u} _{p})\cdot S(\mathbf {v} _{p})\,.

Properties

The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let $H$ be the mean curvature of the surface and $K$ be the Gaussian curvature of the surface, we have

\mathrm {I\!I\!I} -2H\mathrm {I\!I} +K\mathrm {I} =0\,.

As the shape operator is self-adjoint, for $u, v \in T p (M)$ , we find

\mathrm {I\!I\!I} (u,v)=\langle Su,Sv\rangle =\langle u,S^{2}v\rangle =\langle S^{2}u,v\rangle \,.

@@ Line 1: / Line 1: @@
 {{unreferenced|date=October 2015}}
-In [[differential geometry]], the '''third fundamental form''' is a surface metric denoted by <math>\mathbf{III}</math>.  Unlike the [[second fundamental form]], it is independent of the [[surface normal]].
+In [[differential geometry]], the '''third fundamental form''' is a surface metric denoted by <math>\mathrm{I\!I\!I}</math>.  Unlike the [[second fundamental form]], it is independent of the [[surface normal]].
 ==Definition==
-Let <math>S</math> be the [[shape operator]] and <math>M</math> be a [[smooth surface]]. Also, let <math>\mathbf{u}_p</math> and <math>\mathbf{v}_p</math> be elements of the tangent space <math>T_pM</math>. The third fundamental form is then given by
+Let {{mvar|S}} be the [[shape operator]] and {{mvar|M}} be a [[smooth surface]]. Also, let {{math|'''u'''<sub>''p''</sub>}} and {{math|'''v'''<sub>''p''</sub>}} be elements of the tangent space {{math|''T<sub>p</sub>''(''M'')}}. The third fundamental form is then given by
 :<math>
-\mathbf{III}(\mathbf{u}_p,\mathbf{v}_p)=S(\mathbf{u}_p)\cdot S(\mathbf{v}_p)\,.
+\mathrm{I\!I\!I}(\mathbf{u}_p,\mathbf{v}_p)=S(\mathbf{u}_p)\cdot S(\mathbf{v}_p)\,.
 </math>
 ==Properties==
-The third fundamental form is expressible entirely in terms of the [[first fundamental form]] and [[second fundamental form]]. If we let <math>H</math> be the mean curvature of the surface and <math>K</math> be the Gaussian curvature of the surface, we have
+The third fundamental form is expressible entirely in terms of the [[first fundamental form]] and [[second fundamental form]]. If we let {{mvar|H}} be the mean curvature of the surface and {{mvar|K}} be the Gaussian curvature of the surface, we have
 :<math>
-\mathbf{III}-2H\mathbf{II}+K\mathbf{I}=0\,.
+\mathrm{I\!I\!I}-2H\mathrm{I\!I}+K\mathrm{I}=0\,.
 </math>
-As the shape operator is self adjoint, for <math> u,v\in T_p(M)</math>, we find
+As the shape operator is self-adjoint, for {{math|''u'',''v'' ∈ ''T<sub>p</sub>''(''M'')}}, we find
 :<math>
-\mathbf{III}(u,v)=\langle Su,Sv\rangle=\langle u,S^2v\rangle=\langle S^2u,v\rangle
+\mathrm{I\!I\!I}(u,v)=\langle Su,Sv\rangle=\langle u,S^2v\rangle=\langle S^2u,v\rangle\,.
-</math>.
+</math>
 ==See also==

v t e Various notions of curvature defined in differential geometry
Differential geometry of curves	Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature
Differential geometry of surfaces	Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form
Riemannian geometry	Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature
Curvature of connections	Curvature form Torsion tensor Cocurvature Holonomy