Riemannian surface

A Riemann surface is a 2-dimensional compact differentiable surface, together with an infinitesimal element of length (see textbooks on differential and Riemannian geometry, for example, [Nak90]). The curvature K(x) at a point x is the coefficient a in the expansion ... 

In the case of a two-dimensional Riemannian manifold Af with a Riemannian metric gij and with the form of the Riemannian area w = y/det(gij)dx A dy as a symplectic structure (see above), the condition that the group (3 of diffeomorphisms gt preserve the form cj is equivalent to the condition that the domain areas be preserved on the surface when these domains are shifted by the diffeomorphisms gt. Thus, the shifts along integral trajectories of a Hamiltonian field on a two-dimensional symplectic manifold preserve the domain areas. 

In the case of a two-dimensional symplectic manifold, the condition of the locally Hamiltonian character of the field admits another vivid geometrical interpretation. Let gij be a Riemannian metric on and let u) = y/det gij)dx A dy be the form of the Riemannian area. By virtue of the Darboux theorem, one can always choose local coordinates p and q such that the form oj be written in the canonical form dp A dq. Here p and q are certain functions of x and y (and vice versa). Let t be a locally Hamiltonian field t = (i (a , y),Q(x,y)), where P and Q are coordinates of the field in the local system of coordinates p and q. Let us interpret the field v as a velocity field of the flow of liquid of constant density (equal to unity) on the surface M. Let us investigate the variation of the mass of the liquid bounded by an infinitesimal rectangle on the surface when it is shifted along integral trajectories of the field v. It is clear that the mass of this liquid is equal to the area of the rectangle. Therefore, the mass of the liquid contained in a bounded (sufficiently small) domain on is equal to the area of the domain. 

Lemma 5.2.4. There exists a Riemann surface M, a holomorphic mapping x M Cf and a complete Riemannian metric gn on M, such that ... 

Let n = 2x M) > 0. On the manifold M, a function /, holomorphic in the conformal structure on M given by the Riemannian metric T is existent and unique up to multiplication by a constant. This function has simple zeros at the points of the set does not have other zeros, and has poles of order n at infinity. Let M be a Riemann surface of the function y/J and x Af —> M a projection. Then /J is a single-valued function on Af. Define /y/7 formula... 

Setting up a Riemannian metric of class C7 on a compact two-dimensional ori-entable manifold transforms this manifold into a Riemann surface. The charts on it are the local charts of the coordinates isothermic for this metric. The Riemannian metric on the sphere generates a Riemann surface which is homeomorphic... 

The first to discuss the question of constructing nontrivial surfaces with closed geodesics was evidently Darboux [189]. We will say that a Riemannian manifold satisfies the 5C-property if there exists a number / > 0 such that any geodesic on Af is a simple closed curve of length / ( or its multiplicities). 

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