Left-invariant pseudo-Riemannian metrics on four-dimensional lie groups with nonzero Schouten–Weyl tensor

2017 ◽  
Vol 61 (8) ◽  
pp. 81-85
Author(s):  
P. N. Klepikov
Keyword(s):  
Lie Groups ◽  
Weyl Tensor ◽  
Riemannian Metrics

Algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor

2017 ◽  
Vol 95 (1) ◽  
pp. 62-64
Author(s):  
P. N. Klepikov ◽  
E. D. Rodionov
Keyword(s):  
Lie Groups ◽  
Weyl Tensor ◽  
Ricci Solitons

scholarly journals Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

2016 ◽  
Vol 68 (2) ◽  
pp. 669-684 ◽  
Author(s):  
Takahiro HASHINAGA ◽  
Hiroshi TAMARU ◽  
Kazuhiro TERADA
Keyword(s):  
Lie Groups ◽  
Riemannian Metrics ◽  
Invariant Riemannian Metrics

scholarly journals LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 88-95
Author(s):  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Lie Groups ◽  
Complex Structure ◽  
New Left ◽  
Riemannian Metrics ◽  
Nilpotent Lie Groups ◽  
Complex Structures ◽  
Structural Group ◽  
Almost Complex ◽  
Flat Structures ◽  
Simply Connected

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

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