Among other things he came up with the notion of Gaussian curvature.

A sphere has Gaussian curvature at any point on its surface.

At the same time, a plane has zero Gaussian curvature.

The sign of the Gaussian curvature can be used to characterise the surface.

The shape of the intersection is related to the Gaussian curvature.

The sphere and the plane have different Gaussian curvatures, so this is impossible.

Therefore the Gaussian curvature of the geodesic surface under discussion is.

The points at which the Gaussian curvature is zero are called parabolic.

Gaussian curvature is the product of the two principal curvatures.

Continuing in this way we find such independent Gaussian curvatures for a space of n dimensions.

Thus the Gaussian curvature is an intrinsic invariant of a surface.

The pseudosphere is an example of a surface with constant negative Gaussian curvature.

The unit sphere in E has constant Gaussian curvature +1.

The Gaussian curvature of a surface is invariant under local isometry.

It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point.

In particular, the twisted paper model is a developable surface (it has zero Gaussian curvature).

This distance function corresponds to the metric of constant Gaussian curvature +1.

Its relationship both to the Gaussian curvature of two-dimensional surfaces and to tidal forces is discussed.

Asymptotic directions can only occur when the Gaussian curvature is negative (or zero).

Take a hyperbolic plane whose Gaussian curvature is .

In the standard hyperbolic plane (with Gaussian curvature -1 at every point):

For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve.

Then from the definition of Gaussian curvature.

Such a surface has a definite Gaussian curvature K ab.

In particular, the Gaussian curvature is invariant under isometric deformations of the surface.