arXiv subjects

José D. da Silva

Publications and source records attributed to José D. da Silva.

4 recordsLinked to original sources

arXiv2026

Parallelisation of Discrete Exterior Calculus via Representation Theory on Curved and Three-Dimensional Meshes

We establish a universal block-diagonalization framework for Discrete Exterior Calculus (DEC) operators on symmetric meshes, enabling embarrassingly parallel solvers with provable FLOP reductions. We prove that the two fundamental DEC operators, the discrete exterior derivative $d$ and the Hodge star $\star$, are equivariant under isometric finite group actions on simplicial complexes. The proof exploits the permutation representation induced on cochain spaces by the group action. As a consequence, any operator assembled from $d$ and $\star$ (including the Hodge Laplacian, the codifferential, Maxwell-type operators, and elasticity operators) inherits a block-diagonal structure in a single symmetry-adapted basis, which is computed only once per mesh. Unlike spectral methods restricted to flat Platonic domains, the framework applies natively to curved manifolds and is applicable in principle to computational electromagnetism and geometric fluid simulation on symmetric domains. Numerical experiments on a geodesic sphere ($I_h$ symmetry) and a hexagonal torus ($D_{6h}$ symmetry) yield FLOP-based parallel speedups, relative to a dense direct factorization, of up to $62\times$ and $182\times$, respectively. A further experiment on a body-centred-cubic (BCC) tessellation of the flat 3-torus $T^3$ with $T_d$ symmetry confirms equivariance of the exterior derivative, Hodge star, and Hodge Laplacian at machine precision for form degrees $k=0,1,2$ across three mesh resolutions. The FLOP-based sequential speedup approaches its theoretical asymptote of $\approx 9.07\times$, which a standard Schur-multiplicity reduction deepens by a further factor of order $|G|$. These results show that a single symmetry-adapted basis reduces the linear-solve cost of structure-preserving DEC computations on curved and three-dimensional meshes.

math.NA↗

arXiv2024

Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are consequences of the specificities of the ambient manifold and what could be generic. In this work, we propose a theoretical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices; we prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux vector field of the directrix. We characterize rectifying surfaces of constant angle; in other words, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, then the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction.

math.DG↗

arXiv2022

Characterization of manifolds of constant curvature by ruled surfaces

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.

math.DG↗

arXiv2019

Characterization of manifolds of constant curvature by spherical curves

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, the given manifold has constant sectional curvature. We also furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of three-dimensional manifolds. Finally, we also show that the same results are valid for semi-Riemannian manifolds of constant sectional curvature.

math.DG↗