arXiv research

A new metric based on hitting probabilities for directed graphs and Markov chains.

Paper studies metric ribbon graphs and provides a recursion for their volumes.

The paper predicts edge weights in weighted directed networks using metric geometry.

We develop some results on the positivity of direct image bundles in the particular case of a trivial fibration over a one-dimensional base. We also apply the results to study variations of Kahler metrics.

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through s…

We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixe…

PGD-trained models have a preferential direction in their gradients, which improves robustness.

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set K in these directions remain the same straight lines on exit. For example in the…

This note demonstrates how both the concept of distance and the concept of holonomy can be constructed from a suitable network with directed edges (and no lengths). The number of different edge types depends on the signature of the metric and the dimension of the holonomy group. If the holonomy group is of dimension on…

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) an…

Optimizes CNNs by directing gradients along output channels.

For one-parameter degenerations of compact Kähler manifolds, we determine the asymptotic behavior of the first Chern form of the direct image of a Nakano semi-positive vector bundle twisted by the relative canonical bundle, when the direct image is equipped with the L2-metric.

Paper proves direct image sheaf positivity for certain Kähler fibrations.

Constructs a convex Finsler metric on vector bundles under specific conditions.

The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.

We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, have the standard metric $g\_0$ for critic point, althoug…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…

A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of…

Study fixed angle inverse scattering with Riemannian metrics, proving uniqueness and stability.

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to show that this metric is strictly positive everywhere and give estimates. The dire…

New conformally Einstein metrics on Heisenberg group found.

Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\mathbb{R}^…

New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.

Characterizes non-degenerate cyclic metric Lie algebras.

We shall show that $q$-semipositivity of the vector bundle $E$ over a Kähler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on t…

Holomorphic families yield metrics with explicit curvature formulas.

In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used…

Geodesic flows with specific integrals are linked to special 4-webs.

We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < ε(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $ε(x_i)$ varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whethe…

Study of hyperbolic directions in convex projective geometry.

Introduces MSW distances to improve SW metrics.

Directly proves Wu's theorem on negative curvature metrics.

Metric spaces uniquely split into Hilbert and non-line-split parts.

Enhances interpretability of linear latent spaces through automated clustering and ranking.

We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.

ZDP detects drift in large language models without labels, proving key theorems and metrics.

The study examines biharmonic hypersurfaces in Sasakian space forms.

Modified Wasserstein metric for Gaussian distributions, invariant to isometries.

We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…

We show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\h$ of $\g$ always has some negative sectional curvature, unless the semi-simple part of $\h$ is an ideal of $\g$.

In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manif…

The paper studies curvature properties of direct image bundles.

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the…

Paper tackles imbalanced binary classification by optimizing precision and recall directly.

Survey on Chern-Ricci flow for complex manifolds.

Study reconstructs Riemannian metric from Cherenkov radiation in complex media.

A new geometric concept, the dead direction, bridges singular learning theory and information geometry.