§ Gauss, normals, fundamental forms [TODO ]
- consider a parametrization
- at a point on the surface, the tangent vectors are and similarly .
- Let . Then is the first fundamental form . Computed as . Write this as . These numbers depend on the point , or equally, depend on the point .
- Further, we also have a normal vector to the tangent plane. is the unit normal pointing outwards. We can describe it in terms of a parametrization as .
- Gauss map / Gauss Rodrigues map ( ): map from the surface to . sends a point to the unit normal at .
- The tangent plane to on the sphere is parallel to the tanent plane on the surface at , since the normals are the same, as that is the action of which sends the normal at the surface to a point of the sphere / normal to the sphere.
- Thus, the the derivative intuitively "preserves" tangent planes! [as normal directions are determined ].
- If we now think of , it's a map from to . Thus it is a map to the tangent space to itself .
- In terms of this, gauss realized that gaussian curvature is the determinant of the map [ie, the jacobian ]. Curvature is the distortion of areas by the normal. So we can think of it as the ratio of areas
area of image/area of preimage.