The class of Petrov type I curvature tensors is further divided into those for which the span set distinct principal null directions has dimension four (maximally spanning I) or three (nonmaximally I). Explicit examples are provided both vacuum and nonvacuum spacetimes.

According to the problem that how to increase the fidelity of simplified model, this paper put forward a method which using vertex clustering based on principal curvature. Our measure taking the gradual simplify thinking and simplifying the mesh model successive by quadric error metric and principal curvature based on integral invariants. We improve the defect of error accumulation which consis...

Meshes with planar quadrilateral faces are desirable discrete surface representations for architecture. The present paper introduces new classes of planar quad meshes, which discretize principal curvature lines of surfaces in so-called isotropic 3-space. Like their Euclidean counterparts, these isotropic principal meshes meshes are visually expressing fundamental shape characteristics and they ...

Consider oriented surfaces immersed in R. Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k1, k2 is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the ...

Seismic curvature gradient is a useful third-order geometric attribute that has demonstrated its value for subsurface structure analysis, especially fracture detection in fractured reservoirs. However, most recent efforts have been primarily focused on predicting fracture intensity by this attribute; and little has been published on applying it to predict most likely fracture orientation, which...

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces ...