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Distance: Euclidean Geometry
In the third century BC, EuclidEuclid compiled the mathematical knowledge of the time in his book “The Elements”. In his geometrical considerations,...
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Optimal Control, Contact Dynamics and Herglotz Variational Problem
In this paper, we combine two main topics in mechanics and optimal control theory: contact Hamiltonian systems and Pontryagin maximum principle. As...
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Independent component analysis in the light of Information Geometry
I recall my first encounter with Professor Shun-ichi Amari who, once upon a time in Las Vegas, gave me a precious hint about connecting independent...
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Contact Geometry in Optimal Control of Thermodynamic Processes for Gases
AbstractWe solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in...
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Geometry: From Disorder to Order
The axiomatic Euclidean geometry was unique for 2000 years. Then, in the nineteenth century a certain modernity was established with the flourishing...
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Flows of G2-structures on contact Calabi–Yau 7-manifolds
We study the Laplacian flow and coflow on contact Calabi–Yau 7-manifolds. We show that the natural initial condition leads to an ancient solution of...
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Geometry in the Twentieth Century: A Return to Euclid – The Work of Herbert Busemann
This is an exposition of the work of Herbert Busemann (1905–1994), seen as a return to the geometry of Ancient Greece. The importance of this work,...
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Legendrian Cone Structures and Contact Prolongations
We study a cone structure \({\mathcal C} \subset {\mathbb P} D\)...
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Geometry of vectorial martingale optimal transportations and duality
The theory of Optimal Transport and Martingale Optimal Transport (MOT) were inspired by problems in economics and finance and have flourished over...
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Slant Lightlike Submanifolds of Indefinite Contact Manifolds
The geometry of submanifolds with degenerate (lightlike) metric is difficult and strikingly different from the geometry of submanifolds with...
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Geodesic Distance and Monge—Ampère Measures on Contact Sets
We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef...
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Singularities, the Space of Arcs and Applications to Birational Geometry
This paper is an introduction to the space of arcs and the space of jets of an algebraic variety. We also introduce the Nash problem on arc families,...
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Bourgeois contact structures: Tightness, fillability and applications
Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact...
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\(\varepsilon \,\) -Contact Structures and Six-Dimensional Supergravity
We introduceMurcia, A.J. the concept of \(\varepsilon \,\) -contact metric...
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Basic Results of Integral Geometry
The results in this chapter are of a geometric nature, and they emanate from the basic concept of motion-invariant measure associated with a mobile...
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Minkowski Geometry—Some Concepts and Recent Developments
The geometry of finite-dimensional normed spaces (= Minkowski geometry) is a research topic which is related to many other fields, such as convex...
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The Aubry Set and Mather Set in the Embedded Contact Hamiltonian System
We embed the Aubry set and Mather set in the Tonelli Hamiltonian system to the contact Hamiltonian system. We find the embedded Aubry set is the set...