Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature

We review two criteria which have been used to predict the onset of large scale stochasticity in Hamiltonian systems. We show that one of them, due to Toda and based on a local stability analysis of the equations of motion, is inconclusive. An approach based on the local Riemannian curvature K of trajectories correctly predicts chaos if K < 0 everywhere, but�no further conclusions can be drawn. New (counter-)examples are provided.

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REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788714000366 ◽

2014 ◽

Vol 97 (3) ◽

pp. 365-382 ◽

Cited By ~ 1

Author(s):

MOHAMMED LARBI LABBI

Keyword(s):

Riemannian Curvature ◽

Conformally Flat ◽

Flat Manifold ◽

Equality Case ◽

Conformally Flat Manifold

AbstractWe use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{\rm th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.

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Riemannian curvature on the group of area-preserving diffeomorphisms (motions of fluid) of 2-sphere

Physica D Nonlinear Phenomena ◽

10.1016/s0167-2789(96)00192-3 ◽

1997 ◽

Vol 100 (3-4) ◽

pp. 377-389 ◽

Cited By ~ 4

Author(s):

Kyo Yoshida

Keyword(s):

Riemannian Curvature ◽

Area Preserving

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A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

Glasgow Mathematical Journal ◽

10.1017/s0017089512000225 ◽

2012 ◽

Vol 54 (3) ◽

pp. 637-645 ◽

Cited By ~ 1

Author(s):

XIAOHUAN MO ◽

ZHONGMIN SHEN ◽

HUAIFU LIU

Keyword(s):

Scalar Curvature ◽

Simple Proof ◽

Finsler Geometry ◽

Finsler Manifold ◽

Flag Curvature ◽

Finsler Metric ◽

Finsler Metrics ◽

Riemannian Curvature

AbstractIn this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

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