General Nijenhuis tensor: an example of a secondary invariant

Studený, Václav. "General Nijenhuis tensor: an example of a secondary invariant." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1996. [133]-141. <http://eudml.org/doc/220464>.

@inProceedings{Studený1996,
abstract = {The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha $, $\beta $ a mapping \[\langle \alpha , \beta \rangle : (\xi , \zeta ) \mapsto [\alpha (\xi ), \beta (\zeta )] + \alpha \circ \beta ([\xi , \zeta ]) - \alpha ([\xi , \beta (\zeta )]) - \beta ([\alpha (\xi ), \zeta )]),\] where $\xi $, $\zeta $ are vector fields. Then $\langle \alpha , \beta \rangle $ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha , \beta ] = 0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the above Nijenhuis tensor and gives a complete list of all secondary invariants of similar type. In this way he proves that all bilinear natural operators transforming commuting pairs of type (1,1) tensor fields to type (2,1)!},
author = {Studený, Václav},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Proceedings; Geometry; Physics; Winter school; Srní(Czech Republic)},
location = {Palermo},
pages = {[133]-141},
publisher = {Circolo Matematico di Palermo},
title = {General Nijenhuis tensor: an example of a secondary invariant},
url = {http://eudml.org/doc/220464},
year = {1996},
}

TY - CLSWK
AU - Studený, Václav
TI - General Nijenhuis tensor: an example of a secondary invariant
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1996
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [133]
EP - 141
AB - The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha $, $\beta $ a mapping \[\langle \alpha , \beta \rangle : (\xi , \zeta ) \mapsto [\alpha (\xi ), \beta (\zeta )] + \alpha \circ \beta ([\xi , \zeta ]) - \alpha ([\xi , \beta (\zeta )]) - \beta ([\alpha (\xi ), \zeta )]),\] where $\xi $, $\zeta $ are vector fields. Then $\langle \alpha , \beta \rangle $ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha , \beta ] = 0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the above Nijenhuis tensor and gives a complete list of all secondary invariants of similar type. In this way he proves that all bilinear natural operators transforming commuting pairs of type (1,1) tensor fields to type (2,1)!
KW - Proceedings; Geometry; Physics; Winter school; Srní(Czech Republic)
UR - http://eudml.org/doc/220464
ER -