Differential Geometry of Surfaces - Global Differential Geometry of Surfaces

Global Differential Geometry of Surfaces

Although the characterisation of curvature involves only the local geometry of a surface, there are important global aspects such as the Gauss-Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important aspects of the global geometry of surfaces. These include:

• Injectivity radius, defined as the largest r such that two points at a distance less than r are joined by a unique geodesic. Wilhelm Klingenberg proved in 1959 that the injectivity radius of a closed surface is bounded below by the minimum of and the length of its smallest closed geodesic. This improved a theorem of Bonnet who showed in 1855 that the diameter of a closed surface of positive Gaussian curvature is always bounded above by δ; in other words a geodesic realising the metric distance between two points cannot have length greater than δ.

• Rigidity. In 1927 Cohn-Vossen proved that two ovaloids – closed surfaces with positive Gaussian curvature – that are isometric are necessarily congruent by an isometry of E3. Moreover a closed embedded surface with positive Gaussian curvature and constant mean curvature is necessarily a sphere; likewise a closed embedded surface of constant Gaussian curvature must be a sphere (Liebmann 1899). Heinz Hopf showed in 1950 that a closed embedded surface with constant mean curvature and genus 0, i.e. homeomorphic to a sphere, is necessarily a sphere; five years later Alexandrov removed the topological assumption. In the 1980s, Wente constructed immersed tori of constant mean curvature in Euclidean 3-space.

• Carathéodory conjecture: This conjecture states that a closed convex three times differentiable surface admits at least two umbilic points. The first work on this conjecture was in 1924 by Hans Hamburger, who noted that it follows from the following stronger claim : the half-integer valued index of the principal curvature foliation of an isolated umbilic is at most one. The contribution of Hamburger and those of subsequent authors to proving this local conjecture are inconclusive.

• Zero Gaussian curvature: a complete surface in E3 with zero Gaussian curvature must be a cylinder or a plane.

• Hilbert's theorem (1901): no complete surface with constant negative curvature can be immersed isometrically in E3.

• The Willmore conjecture. This conjecture states that the integral of the square of the mean curvature of a torus immersed in E3 should be bounded below by 2 π2. The conjecture has been proved for large classes of torus immersions. It is also known that the integral is a conformal invariant.

• Isoperimetric inequalities. In 1939 Schmidt proved that the classical isoperimetric inequality for curves in the Euclidean plane is also valid on the sphere or in the hyperbolic plane: namely he showed that among all closed curves bounding a domain of fixed area, the perimeter is minimized by when the curve is a circle for the metric. In one dimension higher, it is known that among all closed surfaces in E3 arising as the boundary of a bounded domain of unit volume, the surface area is minimized for a Euclidean ball.

• Systolic inequalities for curves on surfaces. Given a closed surface, its systole is defined to be the smallest length of any non-contractible closed curve on the surface. In 1949 Loewner proved a torus inequality for metrics on the torus, namely that the area of the torus over the square of its systole is bounded below by, with equality in the flat (constant curvature) case. A similar result is given by Pu's inequality for the real projective plane from 1952, with a lower bound of 2/π also attained in the constant curvature case. For the Klein bottle, Blatter and Bavard later obtained a lower bound of . For a closed surface of genus g, Hebda and Burago showed that the ratio is bounded below by 1/2. Three years later Mikhail Gromov found a lower bound given by a constant times g1/2, although this is not optimal. Asymptotically sharp upper and lower bounds given by constants times g/(log g)2 are due to Gromov and Buser-Sarnak, and can be found in Katz (2007). There is also a version for metrics on the sphere, taking for the systole the length of the smallest closed geodesic. Gromov conjectured a lower bound of in 1980: the best result so far is the lower bound of 1/8 obtained by Regina Rotman in 2006.