(Q1305766)

English

Poincaré–Hopf theorem

theorem relating the Euler characteristic of a closed manifold to the number of zeros of a vector field on it

Poincaré–Hopf index formula
Poincaré–Hopf index theorem
Hopf index theorem

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Statements

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\sum _{x\in M \atop v(x)=0}\deg _{\partial \operatorname {ball} (x,\epsilon )\to \mathbb {S} ^{\dim M-1}}\left(y\mapsto {\frac {v(y)}{\|v(y)\|}}\right)=\chi (M)

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in defining formula

M

symbol represents

closed manifold

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v

symbol represents

vector field

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\chi (-)

symbol represents

Euler characteristic

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\deg(-)

symbol represents

degree of a continuous mapping

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\mathbb {S} ^{n}

symbol represents

n-sphere

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x

symbol represents

zero of a function

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v(y)/\|v(y)\|

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1926

1 reference

stated in

Encyclopedia of Mathematics

reference URL

http://www.encyclopediaofmath.org/index.php/Poincaré-Hopf_theorem

title

Poincaré-Hopf theorem (English)

maintained by WikiProject

WikiProject Mathematics

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Identifiers

Encyclopedia of Mathematics article ID

Poincaré-Hopf_theorem

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1 reference

28 October 2013

Microsoft Academic ID

2780799127

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nLab ID

Poincaré–Hopf theorem

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Sitelinks

Wikipedia(11 entries)

cawiki Teorema de Poincaré-Hopf
dewiki Satz von Poincaré-Hopf
enwiki Poincaré–Hopf theorem
frwiki Théorème de Poincaré-Hopf
itwiki Teorema di Poincaré-Hopf
jawiki ポアンカレ・ホップの定理
kowiki 푸앵카레-호프 정리
plwiki Twierdzenie Poincarégo-Hopfa
ruwiki Теорема Пуанкаре о векторном поле
ukwiki Теорема Пуанкаре про векторне поле
zhwiki 庞加莱-霍普夫定理

(Q1305766)

Poincaré–Hopf theorem

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Wikipedia(11 entries)

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