We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of sections of a fibred manifold.
We classify holomorphic isometric actions on complex space forms all whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces.
Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly Kähler $\mathbb{S}^6(1)$
本文研究齐次近Kähler $6$维单位球面$\mathbb{S}^6(1)$的拉格朗日子流形.作为主要结果，我们得到了$\mathbb{S}^6(1)$的紧致拉格朗日子流形的第第二基本型的Simons型积分不等式.此外，我们还证明了等号出现的充要条件是拉格朗日子流形是全测地的$\mathbb{S}^3(1)$或Dillen-Verstraelen-Vrancken的Berger球面$S^3$（见J Math Soc Japan，42：565-584，1990）。
A second-order generalization of the fundamental Lepage form of geometric calculus of variations over fibered manifolds with 2-dimensional base is described by means of insisting on (i) equivalence relation "Lepage differential 2-form is closed if and only if the associated Lagrangian is trivial" and, (ii) the principal component of Lepage form, extending the well-known Poincaré-Cartan form, preserves order prescribed by a given Lagrangian. This approach completes several attempts of finding a Lepage equivalent of a second-order Lagrangian possessing condition (i), which is well-known for firs…
In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge-Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space $M$ is a manifold where a group $G$ acts transitively. Such a space can be understood as a quotient $M \cong G/H$, where $H$ a closed Lie subgroup, is the isotropy group of each point of $M$. The Lie algebra of $G$ may be decomposed into $\mathfrak{g} = \mathfrak{m} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra that generates $H$ and $\mathfrak{m}$ is a subspace. Thus, varia…
