EINSTEIN EXTENSIONS OF RIEMANNIAN MANIFOLDS

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
EINSTEIN EXTENSIONS OF RIEMANNIAN MANIFOLDS
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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