Around Efimov’s differential test for homeomorphism

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Around Efimov’s differential test for homeomorphism. / Alexandrov, Victor.

In: Beitrage zur Algebra und Geometrie, Vol. 62, No. 1, 03.2021, p. 7-20.

Research output: Contribution to journal › Article › peer-review

Harvard

Alexandrov, V 2021, 'Around Efimov’s differential test for homeomorphism', Beitrage zur Algebra und Geometrie, vol. 62, no. 1, pp. 7-20. https://doi.org/10.1007/s13366-020-00534-3

APA

Alexandrov, V. (2021). Around Efimov’s differential test for homeomorphism. Beitrage zur Algebra und Geometrie, 62(1), 7-20. https://doi.org/10.1007/s13366-020-00534-3

Vancouver

Alexandrov V. Around Efimov’s differential test for homeomorphism. Beitrage zur Algebra und Geometrie. 2021 Mar;62(1):7-20. Epub 2020 Nov 20. doi: 10.1007/s13366-020-00534-3

Author

Alexandrov, Victor. / Around Efimov’s differential test for homeomorphism. In: Beitrage zur Algebra und Geometrie. 2021 ; Vol. 62, No. 1. pp. 7-20.

BibTeX

@article{2672b2a3e9144834b8ac3d64a17abba2,

title = "Around Efimov{\textquoteright}s differential test for homeomorphism",

abstract = "In 1968, Efimov proved the following remarkable theorem: Letf: R2→ R2∈ C1be such thatdet f′(x) < 0 for allx∈ R2and let there exist a function a(x) > 0 and constantsC1⩾ 0 , C2⩾ 0 such that the inequalities| 1 / a(x) - 1 / a(y) | ⩽ C1| x- y| + C2and| det f′(x) | ⩾ a(x) | curl f(x) | + a2(x) hold true for allx, y∈ R2. Thenf(R2) is a convex domain andf mapsR2ontof(R2) homeomorphically. Here curl f(x) stands for the curl of f at x∈ R2. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.",

keywords = "diffeomorphism, Efimov{\textquoteright}s theorem, Euclidean 3-space, Gauss curvature, Global asymptotic stability of a dynamical system, Immersed surface, Jacobian conjecture, Milnor{\textquoteright}s conjecture, Riemannian metric, Efimov's theorem, THEOREM, INJECTIVITY, VECTOR-FIELDS, GLOBAL ASYMPTOTIC STABILITY, UMBILICAL POINTS, MAPS, Milnor's conjecture, EXTRINSIC CURVATURE, INITIAL DATA, COMPLETE-SURFACES",

author = "Victor Alexandrov",

year = "2021",

month = mar,

doi = "10.1007/s13366-020-00534-3",

language = "English",

volume = "62",

pages = "7--20",

journal = "Beitrage zur Algebra und Geometrie",

issn = "0138-4821",

publisher = "Springer Berlin",

number = "1",

}

RIS

TY - JOUR

T1 - Around Efimov’s differential test for homeomorphism

AU - Alexandrov, Victor

PY - 2021/3

Y1 - 2021/3

N2 - In 1968, Efimov proved the following remarkable theorem: Letf: R2→ R2∈ C1be such thatdet f′(x) < 0 for allx∈ R2and let there exist a function a(x) > 0 and constantsC1⩾ 0 , C2⩾ 0 such that the inequalities| 1 / a(x) - 1 / a(y) | ⩽ C1| x- y| + C2and| det f′(x) | ⩾ a(x) | curl f(x) | + a2(x) hold true for allx, y∈ R2. Thenf(R2) is a convex domain andf mapsR2ontof(R2) homeomorphically. Here curl f(x) stands for the curl of f at x∈ R2. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.

AB - In 1968, Efimov proved the following remarkable theorem: Letf: R2→ R2∈ C1be such thatdet f′(x) < 0 for allx∈ R2and let there exist a function a(x) > 0 and constantsC1⩾ 0 , C2⩾ 0 such that the inequalities| 1 / a(x) - 1 / a(y) | ⩽ C1| x- y| + C2and| det f′(x) | ⩾ a(x) | curl f(x) | + a2(x) hold true for allx, y∈ R2. Thenf(R2) is a convex domain andf mapsR2ontof(R2) homeomorphically. Here curl f(x) stands for the curl of f at x∈ R2. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.

KW - diffeomorphism

KW - Efimov’s theorem

KW - Euclidean 3-space

KW - Gauss curvature

KW - Global asymptotic stability of a dynamical system

KW - Immersed surface

KW - Jacobian conjecture

KW - Milnor’s conjecture

KW - Riemannian metric

KW - Efimov's theorem

KW - THEOREM

KW - INJECTIVITY

KW - VECTOR-FIELDS

KW - GLOBAL ASYMPTOTIC STABILITY

KW - UMBILICAL POINTS

KW - MAPS

KW - Milnor's conjecture

KW - EXTRINSIC CURVATURE

KW - INITIAL DATA

KW - COMPLETE-SURFACES

UR - http://www.scopus.com/inward/record.url?scp=85096397474&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/ffd28beb-ac26-3743-8d00-62cf2c870329/

U2 - 10.1007/s13366-020-00534-3

DO - 10.1007/s13366-020-00534-3

M3 - Article

AN - SCOPUS:85096397474

VL - 62

SP - 7

EP - 20

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

SN - 0138-4821

IS - 1

ER -

ID: 26205997