Riemann manifold

One of the advantages of the hyper-Kahler structure is that one can identify two apparently different complex manifolds with one hyper-Kahler manifold. Namely, a hyper-Kahler manifold X, g, I, J, K) gives two complex manifolds (X,/) and (X, J), which are not isomorphic in general. For example, on a compact Riemann surface, the moduli space of Higgs bundles and the moduli space of flat PGLr(C)-bundles come from one hyper-Kahler manifold, namely moduli space of 2D-self-duality equation (see [36] for detail.)... 

The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. 

We now wish to introduce a still deeper form of geometry as first suggested by Bernhard Riemann (Sidebar 13.2). Riemann s formalism makes possible a distinction between the space of vectors whose metrical relationships are specified by the metric M and an associated linear manifold by which the vectors and metric are parametrized. Let be an element of a linear manifold (in general, having no metric character) that can uniquely identify the state of a collection of metrical objects X). The Riemannian geometry permits the associated metric M to itself be a function of the state,... 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

Next, if there exists an analytic subset Y in M, such that codim K 2, and if U = M Y is a symplectic manifold, then the manifold M itself is also symplectic. This follows from the Riemann continuation theorem for holomorphic forms. 

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... 

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