The Frölicher–Nijenhuis bracket and the geometry of $$G_2$$ G 2 -and Spin(7)-manifolds

2017 ◽  
Vol 197 (2) ◽  
pp. 411-432 ◽  
Author(s):  
Kotaro Kawai ◽  
Hông Vân Lê ◽  
Lorenz Schwachhöfer
Keyword(s):  
Nijenhuis Bracket

Frölicher–Nijenhuis cohomology on G2- and Spin(7)-manifolds

2018 ◽  
Vol 29 (12) ◽  
pp. 1850075
Author(s):  
Kotaro Kawai ◽  
Hông Vân Lê ◽  
Lorenz Schwachhöfer
Keyword(s):  
Riemannian Manifold ◽  
Differential Form ◽  
Kähler Manifold ◽  
Cohomology Groups ◽  
Kahler Manifold ◽  
Partial Description ◽  
Nijenhuis Bracket

In this paper, we show that a parallel differential form [Formula: see text] of even degree on a Riemannian manifold allows to define a natural differential both on [Formula: see text] and [Formula: see text], defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel [Formula: see text]-form on a [Formula: see text]- and [Formula: see text]-manifold, respectively. We calculate the cohomology groups of [Formula: see text] and give a partial description of the cohomology of [Formula: see text].

scholarly journals The Frölicher-Nijenhuis bracket on some functional spaces

1998 ◽  
Vol 68 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Ivan Kolář ◽  
Marco Modungo
Keyword(s):  
Nijenhuis Bracket

Some applications of the Mangiarotti–Modugno formula for tangent-valued forms

2014 ◽  
Vol 11 (07) ◽  
pp. 1460022
Author(s):  
Ivan Kolář
Keyword(s):  
Lie Algebras ◽  
Classical Approach ◽  
Graded Lie Algebras ◽  
Nijenhuis Bracket ◽  
Fibered Manifolds

First, we present a classical approach to the general connections on arbitrary fibered manifolds. Then we compare this approach with the use of the Frölicher–Nijenhuis bracket by Mangiarotti and Modugno [Graded Lie algebras and connections on a fibered space, J. Math. Pures Appl. 63 (1984) 111–120]. Finally, we demonstrate that the latter viewpoint is very efficient in the theory of torsions of connections on Weil bundles.

scholarly journals Lie groupoids and the Frölicher-Nijenhuis bracket

2013 ◽  
Vol 44 (4) ◽  
pp. 709-730 ◽  
Author(s):  
Henrique Bursztyn ◽  
Thiago Drummond
Keyword(s):  
Lie Groupoids ◽  
Nijenhuis Bracket

GRADED POISSON STRUCTURES AND SCHOUTEN–NIJENHUIS BRACKET ON ALMOST COMMUTATIVE ALGEBRAS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250042 ◽  
Author(s):  
FERDINAND NGAKEU
Keyword(s):  
Poisson Bracket ◽  
Abelian Groups ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Commutative Algebras ◽  
Nijenhuis Bracket

We introduce and study the notion of abelian groups graded Schouten–Nijenhuis bracket on almost commutative algebras and show that any Poisson bracket on such algebras is defined by a graded bivector as in the classical Poisson manifolds. As a particular example, we introduce and study symplectic structures on almost commutative algebras. Our result is a generalization of the ℤ2-graded (super)-Poisson structures.

scholarly journals The Z2-graded Schouten–Nijenhuis bracket and generalized super-Poisson structures

1997 ◽  
Vol 38 (7) ◽  
pp. 3735-3749 ◽  
Author(s):  
J. A. de Azcárraga ◽  
J. M. Izquierdo ◽  
A. M. Perelomov ◽  
J. C. Pérez-Bueno
Keyword(s):  
Poisson Structures ◽  
Nijenhuis Bracket

The Schouten-Nijenhuis bracket and interior products

1997 ◽  
Vol 23 (3-4) ◽  
pp. 350-359 ◽  
Author(s):  
Charles-Michel Marle
Keyword(s):  
Nijenhuis Bracket
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