Cartan's Theorem Research Articles

Unstability problem of real analytic maps

AbstractAs well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

Characterization of coincidence site lattices generated by the composition of simple reflections of cubic lattices

This work provides a comprehensive characterization of coincidence site lattices that arise from the compositions of the reflections of cubic lattices. For this purpose, we apply Cartan's theorem and use reflections as basic isometries, which allows the use of Clifford algebra as a powerful tool to handle reflections and products of reflections, leading to explicit expressions for coincidence indices and the basis of coincidence lattices. The three possible cases were studied: simple reflections, the product of two reflections (rotations), and the product of three reflections, where a factorization that produces improper rotations was provided. In the case of rotations, it is formally proven that Ranganathan's well‐known generating function of coincidence indices for cubic systems under rotations yields all possible coincidence lattices. Overall, this study provides valuable insights into the mathematical properties of coincidence site lattices and the use of Clifford algebras to describe orthogonal transformations.

Rigidity, counting and equidistribution of quaternionic Cartan chains

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

Infinitesimal homogeneity and bundles

Let $$Q\rightarrow M$$ be a principal G-bundle, and $$B_0$$ a connection on Q. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle $$Q\times _GV$$ with respect to $$B_0$$ , and, inspired by the well known Ambrose–Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose–Singer type theorems by an appropriate choice of the initial system of data. We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors. Finally, we introduce natural local homogeneity and local symmetry conditions for triples consisting of a Riemannian metric on M, a principal bundle on M, and a connection on P. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose–Singer and Cartan theorem.

Results on algebroid functions in k-punctured complex plane

The main purpose of this paper is to investigate some properties of algebroid function in the k-punctured complex plane. We establish the first and second fundamental theorems, the Cartan theorem for algebroid function in the k-punctured complex plane, which extend the Nevanlinna theory for algebroid functions from single connected domain to multiple connected domain.

A closedness theorem and applications in geometry of rational points over Henselian valued fields

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the \L{}ojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, H\"{o}lder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.

Biholomorphic Mappings on Banach Spaces

AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

Clifford Algebra As a Bridge Between Discrete and Continuous Worlds

There are several topics of Clifford algebra where discrete and continuous worlds are deeply intertwined, one for all the Cartan theorem that expresses continuous automorphisms by a succession of reflections, but they thrive separately an do not fit in a comprehensive picture. We will focus on some cases where Clifford algebra manifestly shows discrete features with the hope that shedding light on some details may help a more complete scenario to come out of darkness. In particular we will tackle two themes: the first is the ubiquitous role played by the base two expansion of integers in Clifford algebra: for example the interplay between the signature of the vector space and the ’spinorial chessboard’ or the determination of row and column indices of the isomorphic matrix algebra. The second theme is the contribution that could give the Abelian subalgebra of the idempotents to the solution of the SATisfiability problem, the mother of all combinatorial problems.

Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.

Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel

In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given.

On Cartan's theorem for linear operators

AbstractIn this paper, we describe a second main theorem of holomorphic curves in , of hyper‐order strictly less than 1, that involves a general linear operator . As an application, we derive a truncated second main theorem of degenerate holomorphic curves of hyper‐order strictly less than 1 using Nochka weights.

A cohomological obstruction to the existence of compact Clifford–Klein forms

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq 1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q \geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply H. Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $\epsilon$-families of semisimple symmetric pairs.

Schwarz’s Lemma for Slice Clifford Analysis

It is known that the Schwarz lemma in Clifford analysis does not hold at least in the original form. In this article we find that the situation in slice Clifford analysis is totally different. The sharp Schwarz lemma in slice Clifford analysis holds true in the original form, as well as the Cartan theorem, the Hopf lemma, and the Burns-Krantz theorem. The point of these theorems in slice Clifford analysis is that the results hold for such a map \({f :\mathbb{R}^{m+1} \rightarrow \mathbb{R}_0,m}\) that is not a self map for any m ≥ 2.

On automorphisms of quasi-circular domains fixing the origin

It is known that automorphisms of quasi-circular domains fixing the origin are polynomial mappings. By introducing the so-called resonance order and quasi-resonance order, we provide a uniform upper bound for the degree of such polynomial automorphisms. As a particular consequence of our result, we obtain a generalization of the classical Cartan's theorem for circular domains to the quasi-circular case.

The fundamental theorems of algebroid functions on annuli

An extension of Nevanlinna value distribution theory for algebroid functions on annuli is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Cartan theorem, the first fundamental theorem, the second fundamental theorem, deficient values, and the uniqueness of algebroid functions on annuli are proved.

Curvature-adapted submanifolds of symmetric spaces

Curvature-adapted submanifolds have been extensively studied in complex and quaternionic space forms. This paper extends their study to a wider class of ambient spaces. We generalize Cartan's theorem classifying isoparametric hypersurfaces of spheres to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. We classify those with constant principal curvatures in the octonionic planes. Various classification results for hypersurfaces in complex two-plane Grassmannians are also obtained.

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden–Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie–Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

On Griffiths–Lang's conjecture and related topics

We mainly give a brief survey on some methods and results related to Griffiths–Lang's conjecture on the second main theorem of holomorphic curves type into projective varieties. Also presented is the analogue of Cartan's theorem in number theory by Cartan's method, which is a refined form of Schmidt's subspace theorem.

Applications holomorphes propres d'un domaine quasi-disqué sur un domaine disqué

A result of F. Berteloot and G. Patrizio [F. Berteloot, G. Patrizio, A cartan theorem for proper holomorphic mappings of complete circular domains, Adv. Math. 153 (2000) 342–352] states that if f is a proper holomorphic map between two bounded complete circular domains Ω 1 and Ω 2 in C n + 1 ( n ⩾ 1 ), such that f − 1 { 0 } = { 0 } and such that the principal part f p of the Taylor expansions of f at the origine is nondegenerated i.e. f p − 1 { 0 } = { 0 } , then f = f p . Here, we give a partial extension of this result to the case where f is a nondegenerated proper holomorphic map between a quasi-circular domain Ω 1 and a complete circular domain Ω 2 , which are pseudo-convex but not necessarily bounded. We show that if f and its principal part f p are nondegenerated at the origine, then f p − 1 ( Ω 2 ) = Ω 1 .

Reflections, Rotations, and Pythagorean Numbers

In this article, simple reflections, rotations and the Cartan theorem are handled using Clifford algebras. With this tool we provide a constructive proof of the Cartan theorem and the relationship with Pythagorean numbers is discussed.