Third fundamental form
In differential geometry, the third fundamental form is a surface metric denoted by . 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 up and vp be elements of the tangent space Tp(M). The third fundamental form is then given by
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
As the shape operator is self-adjoint, for u,v ∈ Tp(M), we find
See also
- Metric tensor
- First fundamental form
- Second fundamental form
- Tautological one-form
- v
- t
- e
Various notions of curvature defined in differential geometry
of curves
- Curvature
- Torsion of a curve
- Frenet–Serret formulas
- Radius of curvature (applications)
- Affine curvature
- Total curvature
- Total absolute curvature
of surfaces
- Principal curvatures
- Gaussian curvature
- Mean curvature
- Darboux frame
- Gauss–Codazzi equations
- First fundamental form
- Second fundamental form
- Third fundamental form
This differential geometry-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e