At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a … See more

It is also given by $${\displaystyle K={\frac {{\bigl \langle }(\nabla _{2}\nabla _{1}-\nabla _{1}\nabla _{2})\mathbf {e} _{1},\mathbf {e} _{2}{\bigr \rangle }}{\det g}},}$$ where ∇i = ∇ei is the covariant derivative and g is the metric tensor.
At a point p on a … See more

Theorema egregium
Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature … See more

When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.
When a surface has … See more

The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions from that point. We represent the surface by the See more

The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a See more

• Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. A consequence of Minding's theorem is … See more