We study discrete geodesic foliations of surfaces---foliations whose leaves are all approximately geodesic curves---and develop several new variational algorithms for computing such foliations. Our key insight is a relaxation of vector field integrability in the discrete setting, which allows us to optimize for curl-free unit vector fields that remain well-defined near singularities and robustly recover a scalar function whose gradient is well aligned to these fields. We then connect the physics governing surfaces woven out of thin ribbons to the geometry of geodesic foliations, and present a design and fabrication pipeline for approximating surfaces of arbitrary geometry and topology by triaxially-woven structures, where the ribbon layout is determined by a geodesic foliation on a sixfold branched cover of the input surface. We validate the effectiveness of our pipeline on a variety of simulated and fabricated woven designs, including an example for readers to try at home.

Weaving geodesic foliations

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the further condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We also prove that CMC constant angle hypersurfaces in a product $\mathbb{R}\times N$ are either totally geodesic or cylinders.

Hypersurfaces with a canonical principal direction

We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field

A totally geodesic property of Hopf vector fields

It is well-known that ifis a smooth vector field on a given Rie- mannian manifold M n thennaturally defines a submanifold �(M n ) transverse to the fibers of the tangent bundle TM n with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold N l of TM n (1 ≤ l ≤ n) can be realized locally as the image of a sub- manifold F l of M n under some vector fielddefined along F l . For such images �(F l ), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (� of constant length, � normal to F l , M n of constant curvature, M n a Lie group anda left invariant vector field).

Transverse totally geodesic submanifolds of the tangent bundle

Given a Riemannian manifold 
$$N^n$$

 and 
$${\mathcal{Z}}\in {\mathfrak {X}}(N)$$

, an isometric immersion 
$$f:M^m\rightarrow N^n$$

 is said to have the constant ratio property with respect to 
$${\mathcal{Z}}$$

 either if the tangent component 
$${\mathcal{Z}}^T_f$$

 of 
$${\mathcal{Z}}$$

 vanishes identically or if 
$${\mathcal{Z}}^T_f$$

 vanishes nowhere and the ratio 
$$\Vert {\mathcal{Z}}^\perp _f\Vert /\Vert {\mathcal{Z}}^T_f\Vert$$

 between the lengths of the normal and tangent components of 
$${\mathcal{Z}}$$

 is constant along 
$$M^m$$

. It has the principal direction property with respect to 
$${\mathcal{Z}}$$

 if 
$${\mathcal{Z}}^T_f$$

 is an eigenvector of all shape operators of f at all points of 
$$M^m$$

. In this article, we study isometric immersions 
$$f:M^m\rightarrow N^n$$

 of arbitrary codimension that have either the constant ratio or the principal direction property with respect to distinguished vector fields 
$${\mathcal{Z}}$$

 on space forms, product spaces 
$${\mathbb {S}}^n\times {\mathbb {R}}$$

 and 
$${\mathbb {H}}^n\times {\mathbb {R}}$$

, where 
$${\mathbb {S}}^n$$

 and 
$${\mathbb {H}}^n$$

 are the n-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products 
$$I\times _{\rho }{\mathbb {Q}}_\epsilon ^n$$

 of an open interval 
$$I\subset {\mathbb {R}}$$

 and a space form 
$${\mathbb {Q}}_\epsilon ^n$$

. Starting from the observation that these properties are invariant under conformal changes of the ambient metric, we provide new characterizations and classification results of isometric immersions that satisfy either of those properties, or both of them simultaneously, for several relevant instances of 
$${\mathcal{Z}}$$

 as well as simpler descriptions and proofs of some known ones for particular cases of 
$${\mathcal{Z}}$$

 previously considered by many authors. Our methods also allow us to classify Euclidean submanifolds with the property that the normal components of their position vector fields are parallel with respect to the normal connection, and to give alternative descriptions to those in Chen (J Geom 74(1–2): 61–77, 2002) of Euclidean submanifolds whose tangent or normal components of their position vector fields have constant length.

/pdf/geometry-of-submanifolds-with-respect-to-ambient-vector-1ldiz1ia38.pdf

Geometry of submanifolds with respect to ambient vector fields

We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky (n + 1) space.

On the mean curvature of a unit vector field.

We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic unit vector fields for K = 0 and K = 1 and prove a non-existence result for K 6 0,1. We also found a family �! of vector fields on the hyperbolic 2-plane L 2 of curvature −c 2 which generate foliations on T1L 2 with leaves of constant intrinsic curvature −c 2 and of constant extrinsic curvature − c 2 4 .

On the intrinsic geometry of a unit vector field

We introduce three types of helices in three-dimensional Lie groups with left-invariant metric and give their geometrical description similar to that of Lancret. We generalize the results known for the case of three-dimensional Lie groups with bi-invariant metric.

/pdf/generalized-helices-in-three-dimensional-lie-groups-2lpzfxd550.pdf

Alexander Yampolsky

Papers

On the mean curvature of a unit vector field.

On the intrinsic geometry of a unit vector field

A totally geodesic property of Hopf vector fields

Transverse totally geodesic submanifolds of the tangent bundle

Generalized helices in three-dimensional Lie groups