Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0907.1872

Fermat Reals - Nilpotent Infinitesimals and Infinite Dimensional Spaces

Paolo Giordano
View original source ↗Read source document ↗Export metadata ↓

Abstract

F.: Good morning Hermann, I would like to talk with you about infinitesimals. G.: Tell me Pierre. F.: I'm fed up of all these slanders about my attitude to be non rigorous, so I've started to study nonstandard analysis (NSA) and synthetic differential geometry (SDG). G.: Yes, I've read something ... F.: Ok, no problem about their rigour. But, when I've seen that the sine of an infinite in NSA is infinitely near to a real number I was astonished: what is the intuitive meaning of this number, if any? Then, I've seen that to work in SDG I must learn to work in intuitionistic logic ... You know, I love margins of books, and I don't want to loose too much time, I have many things to do ... G.: In SDG they also say that every infinitesimal is at the same time positive and negative, what is the meaning of all these? And why does the square of a first order infinitesimal equal zero, whereas the product of two first order infinitesimals is not necessarily zero? And do you know that from any single infinitesimal in NSA is possible to construct a non measurable set? Without using the axiom of choice! F.: Yes, I know, I know ... Ok, listen: why cannot we start from standard real functions of one real variable and use ... This work is the ideal continuation of this dialogue: a theory of actual infinitesimals that do not need a background of formal logic to be understood, with a clear intuitive meaning and with non trivial applications to differential geometry of both finite and infinite dimensional spaces.

Explore related subjects

math.DGmath.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paolo Giordano. 2009-07-10. Fermat Reals - Nilpotent Infinitesimals and Infinite Dimensional Spaces. https://arxiv.org/abs/0907.1872

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

math.DG↗
arXiv2026

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(π: X \to Y, ξ)$. In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric $g$ implies K-polystability of $(π: X \to Y, ξ)$, in the case that the Ricci curvature of $g$ decays at infinity. As an application, we give a non-existence result: if $M$ is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space $X$ of the cube root of $K_M$ is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

math.DG↗
arXiv2026

Observações sobre funções potenciais de variedades quase-Einstein não compactas

Neste artigo, estudamos o conjunto de funções potenciais em variedades quase Einstein não compactas. Mostramos que o espaço de todas as funções potenciais positivas em uma variedade tridimensional não compacta quase-Einstein tem dimensão no máximo dois, e que a igualdade vale se e somente se a variedade for isométrica a um produto $B\times\mathbb{R}$, onde $B$ é uma superfície $λ$-Einstein ou um dos exemplos obtidos por L. Berard Bergery e descritos no livro de Besse. Além disso, provamos que qualquer variedade quase-Einstein assintoticamente plana $n$-dimensional com $λ=0$ é necessariamente Ricci-plana.

math.DG↗