Abstract : Let be a 2nd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates. We are going to solve this problem by an applicable method which has been recognized by R. Gardner, and classify them.   

where χ(S) is the Euler characteristic of the surface, B r denotes the geodesic r-ball in Hn(b) and Vol(S 2∩Bb,n r ) Vol(B r ) is the volume growth of the domains S2 ∩B r . A natural question arises in this context: can we prove the finiteness of the topology of a not necessarily minimal surface in a Cartan–Hadamard manifold and, moreover, establish a Chern–Osserman-type inequality for its Eule...

It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras which form the set of all the regular equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical ri...

Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space n =Sp(n+ 1)/Sp(1)×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry on n modelled on (spn+1, sp1 × spn). The integrability structure is shown to be geometrically encoded by a Poisson– Nijenhuis stru...

We perform the non-linear realisation or the coset formulation of the pure N = 4, D = 5 supergravity. We derive the Lie superalgebra which parameterizes a coset map whose induced Cartan-Maurer form produces the bosonic field equations of the pure N = 4, D = 5 supergravity by canonically satisfying the Cartan-Maurer equation. We also obtain the first-order field equations of the theory as a twis...