#### 110 Citations

Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders

- Computer Science, Mathematics
- NeurIPS
- 2019

This work endow VAEs with a Poincare ball model of hyperbolic geometry as a latent space and rigorously derive the necessary methods to work with two main Gaussian generalisations on that space. Expand

On Lobachevsky's trigonometric formulae

- Mathematics
- 2012

We elaborate on some important ideas contained in Lobachevsky's Pangeometry and in some of his other memoirs. The ideas include the following: (1) The trigonometric formulae, which express the… Expand

Did Lobachevsky have a model of his Imaginary geometry

- Mathematics
- 2008

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to… Expand

0 70 20 07 v 1 1 F eb 2 00 7 Harmonic fields on mixed Riemannian-Lorentzian manifolds

- 2007

The extended projective disc is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on this metric can be interpreted as the hodograph image of… Expand

On projective and affine equivalence of sub-Riemannian metrics

- Mathematics
- Geometriae Dedicata
- 2019

Consider a smooth connected manifold M equipped with a bracket generating distribution D. Two sub-Riemannian metrics on (M, D) are said to be projectively (resp. affinely) equivalent if they have the… Expand

Distances and Means of Direct Similarities

- Mathematics, Computer Science
- International Journal of Computer Vision
- 2014

The results show that the new divergences presented here, and their means, are both more effective and faster to compute for this task, and are compared in a real-world application: vote-based, scale-invariant object recognition. Expand

Geodesically equivalent metrics in general relativity

- Mathematics, Physics
- 2012

Abstract We discuss whether it is possible to reconstruct a metric from its nonparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general… Expand

Complete Einstein Metrics are Geodesically Rigid

- Physics, Mathematics
- 2008

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g of nonconstant curvature is geodesically rigid: if any other complete metric $${{\bar{g}}}$$ has the same… Expand

Emergence of Minkowski-Spacetime by Simple Deterministic Graph Rewriting

- Physics
- 2021

The causal set program as well as the Wolfram physics project leave open the problem of how a graph that is a (3+1)-dimensional Minkowskispacetime according to its simple geodesic distances, could be… Expand

Geodesic scattering on hyperboloids and Knörrer’s map

- Physics, Mathematics
- Nonlinearity
- 2021

We use the results of Moser and Knörrer on relations between geodesics on quadrics and solutions of the classical Neumann system to describe explicitly the geodesic scattering on hyperboloids. We… Expand