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We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In particular, such spaces are not locally bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups that are rigid in the sense that their only quasiconformal maps...
As shown by V. Vassilyev [V], singularities of arbitrary Lagrangian mappings of three-folds form no integral characteristic class. We show, nevertheless, that in the pseudooptical case the number of singularities counted with proper signs forms an invariant. We give a topological interpretation of this invariant, and its applications. The results of the paper may be considered as a 3-dimensional generalization of the results due to V. I. Arnold [A].
Nous montrons qu’une variété riemannienne de dimension , à courbure de Ricci et à courbure sectionnelle majorée, est une sphère dès que la première valeur propre de son laplacien (resp. son diamètre) est suffisamment proche de (resp. de ).
Nous montrons qu’une surface minimale complété, plongée dans , de courbure totale finie et homéomorphe a moins deux points est l’hélicoïde.
Si dimostra l'esistenza di una struttura complessa compatibile globale sulle varietà quaternionali di Hermite-Weyl compatte regolari. Se ne deducono alcune restrizioni sui numeri di Betti.
We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F.
Then using asymptotic expansions we are able to prove some theorems...
In this paper we consider non-compact cylinder-like surfaces called unduloids and study some aspects of their geometry. In particular, making use of a Kenmotsu-type representation of these surfaces, we derive explicit formulas for the lengths and areas of arbitrary segments, along with a formula for the volumes enclosed by them.
On démontre que si le rayon d’injectivité d’une variété riemannienne compacte est égal à , alors le volume de cette variété est supérieur ou égal à celui de la sphère de même dimension et de courbure sectionnelle constante et égale à . L’égalité ne peut se produire que pour cette sphère précise.
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