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Joeri Van der Veken

Publications and source records attributed to Joeri Van der Veken.

At least 19 recordsLinked to original sources
arXiv2026

Warped geometries of Segre-Veronese manifolds

Segre-Veronese manifolds are smooth submanifolds of tensors comprising the partially symmetric rank-1 tensors. We investigate a one-parameter family of warped geometries of Segre-Veronese manifolds, which includes the standard Euclidean geometry. This parameter controls by how much spherical tangent directions are weighted relative to radial tangent directions. We present closed expressions for the exponential map, the logarithmic map, and the intrinsic distance on these warped Segre-Veronese manifolds, which can be computed efficiently numerically. It is shown that Segre-Veronese manifolds are not geodesically connected in the Euclidean geometry, while they are for some values of the warping parameter. The benefits of geodesics connectedness may outweigh using the Euclidean geometry in certain applications. One such application is presented: numerically computing the Riemannian center of mass for averaging rank-1 tensors.

math.NA↗
arXiv2026

Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$

A tractable definition of the homogeneous nearly Kähler structure on $\mathbb{C}P^3$ is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on $\mathbb{C}P^3$, providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly Kähler $\mathbb{C}P^3$, and the nonexistence of Lagrangians with constant sectional curvature is established.

math.DG↗
arXiv2025

Spheres with parallel mean curvature in $\mathbb{S}^2 \times \mathbb{H}^2$

It is known that a surface with parallel mean curvature vector field in a Riemannian product of two surfaces of constant Gaussian curvature carries a holomorphic quadratic differential. In this paper we consider the Riemannian product of a sphere and a hyperbolic plane of opposite Gaussian curvatures and study the parallel mean curvature surfaces for which the differential vanishes. In particular, we classify all parallel mean curvature spheres, for which the differential vanishes for topological reasons.

math.DG↗
arXiv2025

Manifold-valued function approximation from multiple tangent spaces

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that leverages and unifies two prior techniques: (i) approximating a pullback to the tangent space, and (ii) the Riemannian moving least squares method. The core idea of the new scheme is to combine pullbacks to multiple tangent spaces with a weighted Fréchet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction.

math.NA↗
arXiv2024

Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures

We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it denotes the hyperbolic space $\mathbb{H}^3$ if $\varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $\mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $\mathbb{Q}^3\times\mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $\mathbb{Q}^3\times\mathbb{R}$.

math.DG↗
arXiv2024

Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$

We study hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor, including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form, and of totally umbilical hypersurfaces of $\text{Sol}^4_0$. We also give a closed expression for the Riemann curvature tensor of $\text{Sol}^4_0$, using two integrable complex structures.

math.DG↗
arXiv2024

Totally geodesic Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$

In this paper, we study Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. First, we show that they split into four different classes depending on their behaviour with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.

math.DG↗
arXiv2023

Scroll Waves and Filaments in excitable Media of higher spatial Dimension

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of co-dimension two in an N -dimensional space and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have non-local connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.

math.DS↗
arXiv2023

Minimal $δ(2)$-ideal Lagrangian submanifolds and the Quaternionic projective space

We construct an explicit map from a generic minimal $δ(2)$-ideal Lagrangian submanifold of $\mathbb{C}^n$ to the quaternionic projective space $\mathbb{H}P^{n-1}$, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for $n=3$, since the above mentioned map then provides a one-to-one correspondence between minimal $δ(2)$-ideal Lagrangian submanifolds of $\mathbb{C}^3$ and minimal totally complex surfaces in $\mathbb{H}P^2$ which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in $\mathbb{H}P^2$ and minimal Lagrangian surfaces in $\mathbb{C}P^2$.

math.DG↗
arXiv2021

Tensor completion using geodesics on Segre manifolds

We propose a Riemannian conjugate gradient (CG) optimization method for finding low rank approximations of incomplete tensors. Our main contribution consists of an explicit expression of the geodesics on the Segre manifold. These are exploited in our algorithm to perform the retractions. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and identification of pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we recover the tensor decomposition from less than $10\%$ of the data.

math.DG↗
arXiv2021

Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$

The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kähler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space.

math.DG↗
arXiv2020

$H$-umbilical Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$

$H$-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that in the homogeneous nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$, also $H$-umbilical Lagrangian submanifolds are automatically totally geodesic.

math.DG↗
arXiv2020

A geometric study of marginally trapped surfaces in space forms and Robertson-Walker spacetimes -- an overview

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.

math.DG↗
arXiv2020

Lagrangian submanifolds of the complex hyperbolic quadric

We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used to define local angle functions on any Lagrangian submanifold of ${Q^*}^n$. We prove that a Lagrangian immersion into ${Q^*}^n$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagrangian immersion to these principal curvatures. The theorems are illustrated with several examples of spacelike hypersurfaces of anti-de Sitter space and their Gauss maps. Finally, we classify some families of minimal Lagrangian submanifolds of ${Q^*}^n$: those with parallel second fundamental form and those for which the induced sectional curvature is constant. In both cases, the Lagrangian submanifold is forced to be totally geodesic.

math.DG↗
arXiv2019

Rotationally invariant constant Gauss curvature surfaces in Berger spheres

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant $K_0$, which we determine explicitly and depends on the geometry of the ambient Berger sphere. For values of $K_0 \leq K \leq K_P$, for a specific constant $K_P$, it was not known until now whether complete constant Gauss curvature $K$ surfaces existed in Berger spheres, so our classification provides the first examples. For $K > K_P$, we prove that the rotationally invariant spheres from our classification are the only topological spheres with constant Gauss curvature in Berger spheres.

math.DG↗