scholarly journals Non-orthogonalisable vector fields on spheres

1984 ◽  
Vol 27 (3) ◽  
pp. 275-281
Author(s):  
Martin Raussen
Keyword(s):  
Vector Fields ◽  
Fibre Bundle ◽  
Basis Vector ◽  
Image Position

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundlewith fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a mapwhich is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn

On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations

1972 ◽  
Vol 24 (5) ◽  
pp. 799-804 ◽  
Author(s):  
R. L. Bishop ◽  
S.I. Goldberg
Keyword(s):  
Riemannian Manifold ◽  
Covariant Derivative ◽  
Ricci Tensor ◽  
Vector Fields ◽  
Conformally Flat ◽  
Image Position

Let (M, g) be a C∞ Riemannian manifold and A be the field of symmetric endomorphisms corresponding to the Ricci tensor S; that is,We consider a condition weaker than the requirement that A be parallel (▽ A = 0), namely, that the “second exterior covariant derivative” vanish ( ▽x▽YA — ▽Y ▽XA — ▽[X,Y]A = 0), which by the classical interchange formula reduces to the propertywhere R(X, Y) is the curvature transformation determined by the vector fields X and Y.The property (P) is equivalent toTo see this we observe first that a skew symmetric and a symmetric endomorphism commute if and only if their product is skew symmetric.

Perturbation of elliptic operators and complex dynamics of parabolic partial differential equations

2000 ◽  
Vol 130 (2) ◽  
pp. 397-418 ◽  
Author(s):  
M. Prizzi
Keyword(s):  
Parabolic Equation ◽  
Vector Fields ◽  
Complex Dynamics ◽  
Neumann Boundary ◽  
Semilinear Parabolic Equation ◽  
Centre Manifold ◽  
Smooth Bounded Domain ◽  
Smooth Potential ◽  
Image Position ◽  
Semilinear Parabolic

Let Ω ⊂ RN be a smooth bounded domain. Let be a second-order strongly elliptic differential operator with smooth symmetric coefficients. Let B denote the Dirichlet or the Neumann boundary operator. We prove the existence of a smooth potential a : Ω → R such that all sufficiently small vector fields on RN + 1 can be realized on the centre manifold of the semilinear parabolic equation by an appropriate nonlinearity f : ( x, s, w ) ∈ Ω x R x RN ↦ f ( x, s, w ) ∈ R.For N = 2, n, k ∈ N, we prove the existence of a smooth potential a : Ω → R such that all sufficiently small k-jets of vector fields on Rn can be realized on the centre manifold of the semilinear parabolic equation by an appropriate nonlinearity f : ( x, s ) ∈ Ω x R ↦ f (x, s ) ∈ R2 ( here, ‘·’ denotes the scalar product in R2).

The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity

1995 ◽  
Vol 125 (4) ◽  
pp. 675-699 ◽  
Author(s):  
Michael Shearer ◽  
Yadong Yang
Keyword(s):  
Shock Waves ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Mixed Type ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Cubic Nonlinearity ◽  
System Of Conservation Laws ◽  
Admissible Solutions ◽  
Image Position

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation lawswhere σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

Global attraction for systems with almost C1 vector fields

2005 ◽  
Vol 135 (4) ◽  
pp. 815-832
Author(s):  
Zhanyuan Hou
Keyword(s):  
Linear Systems ◽  
Global Attractor ◽  
Differential System ◽  
Vector Fields ◽  
Single Point ◽  
Sufficient Conditions ◽  
Global Attraction ◽  
Almost Everywhere ◽  
Image Position ◽  
Special Case

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

scholarly journals On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

2019 ◽  
pp. 5-20
Author(s):  
Artem Atanov ◽  
Alexander Loboda
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourth Order ◽  
Algebraic Surface ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Basis Vector ◽  
Invariant Polynomial ◽  
Real Hypersurfaces ◽  
Symmetry Algebras

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

scholarly journals Almost Hermitian homogeneous structures

1988 ◽  
Vol 31 (3) ◽  
pp. 375-395 ◽  
Author(s):  
Elsa Abbena ◽  
Sergio Garbiero
Keyword(s):  
Vector Fields ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Almost Complex ◽  
Dimension 2 ◽  
Homogeneous Structures ◽  
Image Position

Let (M, g, J) be an almost Hermitian manifold. More precisely, M is a ∞ differentiable manifold of dimension 2n, J is an almost complex structure on M, i.e. it is a tensor field of type (1, 1) such thatfor any X∈(M), ((M) is the Lie algebra of ∞ vector fields on M), and g is a Riemannian metric compatible with J, i.e.

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