**Remarks on the Schouten-Nijenhuis bracket**

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Postscript Document

Peter Michor

**The paper is published:**
Rendiconti del Circolo Matematico di Palermo, Serie II, , Suppl. 16 (1987), 208--215

**MSC 2000**

- 53A55 Differential invariants (local theory), geometric objects

**Abstract**

A global definition of the Schouten bracket is given, and
a global proof of its connection to Poisson structures
on manifolds. Some properties are derived, in
particular naturality. Then all natural bilinear concomitants
of multi vector fields with the same degree behavior as the
Schouten Nijenhuis bracket are determined: only constant
multiples of the Schouten-Nijenhuis bracket appear.

**Keywords:**
*Schouten-Nijenhuis bracket, Schouten bracket*

**Notes**

The methods of proof were later explained in the book:
Kolar, Ivan; Michor, Peter W.; Slovak, Jan
Natural operations in differential geometry. (English)
Berlin: Springer-Verlag. vi, 434 p. (1993).
The results are cited but not proved.