Isothermal coordinates

Abstract: In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density, and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more than a way to compute a length. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be evaluated continuously on a Riemannian metric space. From one hand, this new continuous framework proves that prescribing a Riemannian metric field is equivalent to the local control in the $\mathbf{L}^1$ norm of the interpolation error. This proves the consistency of classical metric-based mesh adaptation procedures. On the other hand, powerful mathematical tools are available and are well defined on Riemannian metric spaces: calculus of variations, differentiation, optimization$,\dots$, whereas these tools are not defined on discrete meshes.

Abstract: This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute.

Abstract: Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ -convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W 2,2 isometric immersion of a given 2 d metric into .

Abstract: A CR (i.e. partially complex) 5-manifold is contructed as a sphere bundle over an arbitrary 3-manifold with conformal metric. This so-called twistor CR manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract CR manifold to arise via the construction. The above correspondence is then used to prove that a twistor CR manifold is locally imbeddable as a real hypersurface in C3 only if it is real-analytic with respect to a suitable atlas. Introduction. There is a very beautiful interplay between the conformal geometry of Riemannian manifolds and complex analysis, examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self-dual Riemannian 4-manifolds (Penrose [1976], Atiyah et al. [1978]). Intermediating between these two theories is a relationship between 3dimensional conformal geometry and CR (partially complex) manifolds of dimension 5. In this paper, we will explain this relationship, focusing on the imbedding problem: When can these partially complex manifolds be realized as real hypersurfaces in complex 3-manifolds? The idea of associating a CR manifold with a conformal Riemannian 3-manifold originated with Penrose [1975,1983], who, starting with a real-analytic space-like hypersurface in a real-analytic Lorentzian 4-manifold, constructed a real hypersurface in a complex 3-manifold. If the space-like hypersurface is totally geodesic (or merely all-umbilic), Penroses's CR manifold coincides, in our terminology, with the " twistor CR manifold" of the conformal structure induced on the space-like hypersurface. After establishing notation and conventions in ?0, principal results on CR manifolds are reviewed and the notions of CR vector bundle and CR contact form are introduced. In ?1, a general machine for constructing involutive distributions is produced, and in ?2 this construction is used to associate to every conformal Riemannian 3-manifold a CR 5-manifold. The CR manifolds arising in this way are characterized abstractly in ?4 in terms of properties presented in ?3. Then, in ?5, it is shown that these CR manifolds are isomorphic to real hypersurfaces of complex Received by the editors March 15, 1983 and, in revised form, July 8, 1983. 1980 Mathematics Subject Classificcation. Primary 32F25; Secondary 53A30, 83C99. 'Work supported in part by NSF grant MCS-812-0790. 001984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page