ON THE CONTINUITY OF LIAO QUALITATIVE FUNCTIONS OF DIFFERENTIAL SYSTEMS AND APPLICATIONS

2005 ◽  
Vol 07 (06) ◽  
pp. 747-768 ◽  
Author(s):  
XIONGPING DAI
Keyword(s):  
Riemannian Manifold ◽  
Stochastic Stability ◽  
Vector Fields ◽  
Finite Dimension ◽  
Skew Product ◽  
Ergodic System ◽  
Differential Systems ◽  
Lyapunov Spectra ◽  
Skew Product Flows ◽  
Global Linearization

Let 𝔛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let [Formula: see text], 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 𝔛r(M), Liao defined functions [Formula: see text], k = 1, …, ℓ, on [Formula: see text], which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every [Formula: see text] depends Cr-1-continuously upon [Formula: see text] and Cr-continuously on [Formula: see text] for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

scholarly journals On Exponential Dichotomy of Variational Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Bogdan Sasu
Keyword(s):  
Difference Equations ◽  
Exponential Dichotomy ◽  
New Method ◽  
Skew Product ◽  
Skew Product Flows

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

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.

LIAO STYLE NUMBERS OF DIFFERENTIAL SYSTEMS

2004 ◽  
Vol 06 (02) ◽  
pp. 279-299 ◽  
Author(s):  
XIONGPING DAI
Keyword(s):  
Differential System ◽  
Compact Riemannian Manifold ◽  
Ergodic Measure ◽  
Ergodic System ◽  
View Point ◽  
Differential Systems ◽  
Linearly Independent ◽  
Characterization Theorems ◽  
Positive Integers ◽  
Style Number

For any C1 differential system S on a compact Riemannian manifold M of dimension d with d≥2, this paper studies the Liao style numbers, κ(S) (or respectively, κ*(S)) of S from the view-point of ergodic theory. Here κ(S) (κ*(S)) is the largest number of moving vectors (or respectively, conjugate-) of the differential system S that are mean linearly independent. For any ergodic measure ν of S, two positive integers κ*(ν) and κ(ν), called the reduced and non-reduced style number of ν respectively, are introduced. The connection between the style numbers of the system (M,S) and ones of the ergodic system (M,S; ν) are discovered by the variational principle of style number proved in the paper. Several characterization theorems with respect to the style numbers κ*(S), κ*(ν), κ(S) and κ(ν) are presented respectively.

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