Back to Search
Start Over
Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type.
- Source :
-
Nonlinearity . Aug2021, Vol. 34 Issue 8, p1-27. 27p. - Publication Year :
- 2021
-
Abstract
- We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09517715
- Volume :
- 34
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Nonlinearity
- Publication Type :
- Academic Journal
- Accession number :
- 151508984
- Full Text :
- https://doi.org/10.1088/1361-6544/abed39