Translation surface in the Galilean space
... constant Gaussian K and mean curvature H, and translation surfaces that are Weingarten surfaces as well. A translation surface is a surface that can locally be written as the sum of two curves α, β x(u, v) = α(u) + β(v). Translation surfaces in the Euclidean and Minkowski space having constant Gauss ...
... constant Gaussian K and mean curvature H, and translation surfaces that are Weingarten surfaces as well. A translation surface is a surface that can locally be written as the sum of two curves α, β x(u, v) = α(u) + β(v). Translation surfaces in the Euclidean and Minkowski space having constant Gauss ...
4-2 and 4-3
... A line is a transversal if and only if it intersects two or more lines, each at a different point. ...
... A line is a transversal if and only if it intersects two or more lines, each at a different point. ...
5 The hyperbolic plane
... As we see above, the analogy between Euclidean geometry and its theorems and the geometry of the hyperbolic plane is very close, so long as we replace lines by geodesics, and Euclidean isometries (translations, rotations and reflections) by the isometries of H or D. In fact it played an important hi ...
... As we see above, the analogy between Euclidean geometry and its theorems and the geometry of the hyperbolic plane is very close, so long as we replace lines by geodesics, and Euclidean isometries (translations, rotations and reflections) by the isometries of H or D. In fact it played an important hi ...
Riemannian connection on a surface
For the classical approach to the geometry of surfaces, see Differential geometry of surfaces.In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form . These concepts were put in their final form using the language of principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.