Simplices modelled on spaces of constant curvature (original) (raw)

Riemannian simplices and triangulations

Geometriae Dedicata, 2015

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.

On non-Riemannian sectional curvature in Riemannian homogeneous spaces

Tohoku Mathematical Journal, 1971

In this note we give an M. Berger-L. W. Green type inequality [ 2 ] relating non-Riemannian sectional curvature and conjugate points in naturally reductive Riemannian homogeneous spaces (cf [1 3, Chapter X] for all the necessary details). We first recall the result of Berger and Green: Let M be a compact orientable Riemannian manifold of dimension Ξg2, Γ the scalar curvature and Λ/ΛJΊC ,K>0, a lower bound of the distance of any point to its first conjugate point along any geodesic. Then

Degenerate pseudo-Riemannian metrics

In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming from invariant theory. This involves the existence of a boost which is assumed to extend to a neighbourhood. This approach proves to be very fruitful: It produces a class of metrics containing all known examples of I-degenerate metrics. To date, only Kundt and Walker metrics have been given, however, our study gives a plethora of examples showing that I-degenerate metrics extend beyond the Kundt and Walker examples. The approach also gives a useful criterion for a metric to be I-degenerate. Specifically, we use this to study the subclass of VSI and CSI metrics (i.e., spaces where polynomial curvature invariants are all vanishing or constants, respectively).

Local and Global Homogeneity for Manifolds that admit a Positive Curvature Metric

arXiv: Differential Geometry, 2019

In this note we study globally homogeneous Riemannian quotients Gammabackslash(M,ds2)\Gamma\backslash (M,ds^2)Gammabackslash(M,ds2) of homogeneous Riemannian manifolds (M,ds2)(M,ds^2)(M,ds2). The Homogeneity Conjecture is that Gammabackslash(M,ds2)\Gamma\backslash (M,ds^2)Gammabackslash(M,ds2) is (globally) homogeneous if and only if (M,ds2)(M,ds^2)(M,ds2) is homogeneous and every gammainGamma\gamma \in \GammagammainGamma is of constant displacement on (M,ds2)(M,ds^2)(M,ds2). We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition can be dropped.

Curvature measures of pseudo-Riemannian manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal)

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric ( 0 , 2 ) {(0,2)} -tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz–Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz–Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern–Gauss–Bonnet integrand. Finally, we deduce a Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.

Measuring the conformity of non-simplicial elements to an anisotropic metric field

International Journal for Numerical Methods in Engineering, 2005

This paper extends an approach for measuring the element conformity of simplices to non-simplicial elements of any type, in spaces of arbitrary dimension. Element non-conformity is defined as the difference between a given size specification map, in the form of a Riemannian metric tensor, and the actual metric tensor of the element. An approach to the measurement of non-conformity coefficients of non-simplicial elements based on sub-simplex division is proposed. An analysis of the measure's behaviour presented for quadrilaterals, hexahedra, prisms and pyramids shows that the measure is sensitive to size, stretching and orientation variations, as well as to other types of element shape degeneration. Finally, numerical applications show that the metric conformity measure can be used as a quality measure to quantify the discrepancy between a whole non-simplicial mesh and a complex anisotropic size specification map.

Simplicial nonpositive curvature

2006

We introduce a family of conditions on a simplicial complex that we call local k-largeness (k ≥ 6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

A universal measure of the conformity of a mesh with respect to an anisotropic metric field

International Journal for Numerical Methods in Engineering, 2004

In this paper, a method is presented to measure the non-conformity of a mesh with respect to a size specification map given in the form of a Riemannian metric. The measure evaluates the difference between the metric tensor of a simplex of the mesh and the metric tensor specified on the size specification map. This measure is universal because it is a unique, dimensionless number which characterizes either a single simplex of a mesh or a whole mesh, both in size and in shape, be it isotropic or anisotropic, coarse or fine, in a small or a big domain, in two or three dimensions. This measure is important because it can compare any two meshes in order to determine unequivocally which of them is better. Analytical and numerical examples illustrate the behaviour of this measure. 2676 P. LABBÉ ET AL. Vallet [2, showed that a size specification map using a metric tensor representation eased the generation of adapted and anisotropic meshes by combining the desired size and stretching into a single mathematical concept. Metric tensors modify the way distances are measured. The adapted and anisotropic mesh in the real space is constructed by building a regular, isotropic and unitary mesh in the metric tensor space.

Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

The Journal of Geometric Analysis, 2021

The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.