We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of 2N dimensional Ricci flat (pseudo-) Riemannian geometries.

Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces—disc...

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finitedimensional vector space H1(M,R) are stable norms of a Riemannian metric on M . If the dimension of M is at least three, I. Babenko and F. Bala...

In this paper we show that for Riemannian manifolds with boundary the natural restriction map is a quasifibration between spaces of metrics of positive scalar curvature. We apply this result to study homotopy properties of spaces of such metrics on manifolds with boundary.

We construct the moduli space of r−jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratifica-tion of moduli spaces is car...

The connguration spaces of mechanical systems usually support Riemannian metrics which have a explicitly solvable geodesic ows and parallel transport operators. While not of primary interest, such metrics can be used to generate integration algorithms by using the known parallel transport to evolve points in velocity phase space.

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-selfdual Riemannian metrics. The key new ingredient is a proof that the connected sum CP2#CP2#CP2#CP2#CP2 of five reverse-oriented complex projective planes admits such metrics.