We shall describe such maximal atlas and provide it with an algebraic structure that brings along the holonomy groupoid almost for free. This will, at the same time, correct an attempt I made in [5] to ”algebraicize” some considerations of Haefliger. The content is not very original; one may say that it only synthesises insights that are more or less explicit in the work of Haefliger, Pradines,...

Abstract We consider a singular holomorphic vector field in neighborhood of 0∈ C3and suppose that there is foliation codimension one (outside its set, given by decomposition this into complex surfaces, called leaves) to which it tangent. This means that, when both objects are non-singular, the orbits contained leaves foliation. First we desingularizations objects, trying relate their final mode...

Given a foliation F of a closed 3-manifold and a Smale flow φ transverse to F, we associate a “simplest” branched surface with the pair (F, φ), which is unique up to two combinatorial moves. We show that all branched surfaces constructed from F and φ can be obtained from the simplest model by applying a finite sequence of these moves chosen so that each intermediate branched surface also carrie...

The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14],...

Accurate and up-to-date mapping and monitoring of rubber plantations is challenging. In this study, we presented a simple method for rapidly and accurately mapping rubber plantations in the Xishuangbanna region of southwest China using phenology-based vegetation index differencing. Temporal profiles of the Normalized Difference Vegetation Index (NDVI), Enhanced Vegetation Index (EVI), Atmospher...

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature, which permit the momentum constraint to be solved exactly. This family includes, as special cases, the extrinsic time gauges that have been exploited in the pa...