In the last few years, many works have appeared containing examples and general results on harmonicity and minimality of vector fields in different geometrical situations. This survey will be devoted to describe many of the known examples, as well as the general results from where they are obtained.

Unit Vector Fields that are Critical Points of the Volume and of the Energy: Characterization and Examples

We compute the energy of a Frenet vector field and the pseudoangle between Frenet vectors for a given non-null curve C in a semi-Euclidean space of signature (n, ν) It is shown that the energy and pseudoangle can be expressed in terms of the curvature functions of C

On the energy and pseudoangle of Frenet vector fields in R_v^n

In this paper, we construct the energy for the ED-frame field of the first and second kind on an orientable hypersurface in Minkowski space. We obtain the geometric properties of some graphics by way of energy. We apply totally diverse discussion and approach to illustrate bending energy functional for the ED-frame field of the first and second kind. Finally, we have a new relation for angle and energy of the curve on an orientable hypersurface in Minkowski space.

An approach to energy and elastic for curves with extended Darboux frame in Minkowski space

In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.

3-quasi-Sasakian manifolds

We consider a q-dimensional distribution as a section of the Grassmannian bundle Gq(M ) of q-planes and we derive, in terms of the intrinsic torsion of the corresponding S(O(q)×O(n−q))-structure, the conditions that this map must satisfy in order to be critical for the functionals energy and volume. Using this it is shown that invariant Riemannian foliations of homogeneous Riemannian manifolds which are transversally symmetric determine harmonic maps and minimal immersions. In particular, canonical homogeneous fibrations on rank one normal homogeneous spaces or on compact irreducible 3-symmetric spaces provide many examples of harmonic maps and minimal immersions of compact Riemannian manifolds.

https://www.ems-ph.org/journals/show_pdf.php?issn=0213-2230&vol=30&iss=1&rank=12

Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion.

The state of the art in protein structure prediction (PSP) is currently achieved by complex deep learning pipelines that require several input features. In this paper, we demonstrate the relevance of Geometric Algebra (GA) for modelling protein features in PSP. We do so by proposing a novel GA metric based on the relative orientations of amino acid residues. We then employ this metric as an additional input feature to a Graph Transformer (GT) to aid the prediction of the 3D coordinates of a protein. Adding this GA-based orientational information improves the accuracy of the predicted coordinates even after few learning iterations and on a small dataset. 

Using a Graph Transformer Network to Predict 3D Coordinates of Proteins via Geometric Algebra Modelling

The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations \(\) are critical points. Later, we prove the instability for these fibrations.

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

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Using a Graph Transformer Network to Predict 3D Coordinates of Proteins via Geometric Algebra Modelling

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations