EINSTEIN EXTENSIONS OF RIEMANNIAN MANIFOLDS

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F21%3A50018355" target="_blank" >RIV/62690094:18470/21:50018355 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.ams.org/journals/tran/2021-374-09/S0002-9947-2021-08259-8/home.html" target="_blank" >https://www.ams.org/journals/tran/2021-374-09/S0002-9947-2021-08259-8/home.html</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/tran/8259" target="_blank" >10.1090/tran/8259</a>

Jazyk výsledku

angličtina

Název v původním jazyce

EINSTEIN EXTENSIONS OF RIEMANNIAN MANIFOLDS

Popis výsledku v původním jazyce

Given a Riemannian space N of dimension n and a field D of symmetric endomorphisms on N, we define the extension M of N by D to be the Riemannian manifold of dimension n + 1 obtained from N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N and D which imply that the extension M is Einstein. In particular, we show that in this case, D has constant eigenvalues; moreover, they are all integer (up to scaling) if det D not equal 0. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of D in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of D, including the complete classification for the case when D has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kahler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when N is a Lie group with a left-invariant metric, under some additional assumptions.

Název v anglickém jazyce

EINSTEIN EXTENSIONS OF RIEMANNIAN MANIFOLDS

Popis výsledku anglicky

Given a Riemannian space N of dimension n and a field D of symmetric endomorphisms on N, we define the extension M of N by D to be the Riemannian manifold of dimension n + 1 obtained from N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N and D which imply that the extension M is Einstein. In particular, we show that in this case, D has constant eigenvalues; moreover, they are all integer (up to scaling) if det D not equal 0. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of D in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of D, including the complete classification for the case when D has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kahler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when N is a Lie group with a left-invariant metric, under some additional assumptions.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

ISSN

0002-9947

e-ISSN

—

Svazek periodika

374

Číslo periodika v rámci svazku

9

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

25

Strana od-do

6059-6083

Kód UT WoS článku

000687216900002

EID výsledku v databázi Scopus

2-s2.0-85111186047