arXiv subjects

Ruth Paola Moas

Publications and source records attributed to Ruth Paola Moas.

2 recordsLinked to original sources

arXiv2023

The sub-Riemannian geometry of screw motions with constant pitch

We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given lambda in R, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch lambda. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R^7 rtimes SO(7) involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".

math.DG↗

arXiv2021

Harmonic unit normal sections of Grassmannians associated with cross products

Let G(k,n) be the Grassmannian of oriented subspaces of dimension k of R^n with its canonical Riemannian metric. We study the energy of maps assigning to each P \in G(k,n) a unit vector normal to P. They are sections of a sphere bundle E_{k,n}^1 over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E_{k,n}^1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J(P) on P^{\bot} to each P\in G(2,8). We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G(2,8). This generalizes the harmonicity of the canonical almost complex structure of S^6.

math.DG↗