Invariant monotone vector fields on Riemannian manifolds

Various concepts of invariant monotone vector fields on Riemannian manifolds are introduced. Some examples of invariant monotone vector fields are given. Several notions of invexities for functions on Riemannian manifolds are defined and their relations with invariant monotone vector fields are studied.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Killing vector fields on Riemannian and Lorentzian 3-manifolds

We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field T. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature S of g and the function $\text{Ric}(T,T)$, where Ric is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of T. Our classification generalizes that of Sasakian structures, which is the special case when $\text{Ric}(T,T)=2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\text{Ric}(T,T)$, for g to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that T has unit length and the coordinates derived in our classification are globally defined on ${\mathbb{R}}^3$, we give conditions under which S completely determines when the metric will be geodesically complete. In the event that the 3-manifold M is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.     

Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

Let (M,g) be a compact Riemannian manifold and T₁M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from (M,g) to , being the Sasaki metric on T₁M, have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T₁M, are particular examples of g-natural metrics. We equip T₁M with an arbitrary Riemannian g-natural metric , and investigate the harmonicity of a unit vector field V of M, thought as a map from (M,g) to . We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold.     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Flows associated to Cameron-Martin type vector fields on path spaces over a Riemannian manifold

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.     

拟常曲率流形中全脐点子流形的拼嵌定理

We study the global umbilic submanifolds with parallel mean curvature vector fields in a Riemannian manifold with quasi constant curvature and get a local pinching theorem about the length of the second fundamental form.     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.     

An existence result for maximizations with respect to cones

A sufficient condition is given for the existence of a solution to a generalized Pareto maximization problem in which maximization is defined in terms of cones. This result generalizes the fact that an upper semicontinuous real-valued function achieves its maximum on a compact set.     

Measure solutions for impulsive evolution equations with measurable vector fields

In this paper we present some results on the question of existence of generalized or measure valued solutions for semilinear impulsive evolution equations on Banach spaces with the nonlinear parts being merely measurable and bounded on bounded sets. This is a far reaching generalization of the previous results of the author and others. It may be interesting to consider extension of the results of this paper to cover impulsive differential inclusions and their potential applications to control theory and uncertain dynamics.     

Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

We compute the homology of the Lie algebra W ₁ of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W ₁-valued currents on the line.     

Unique normal form of a class of 3 dimensional vector fields with symmetries

We present the unique normal form of a class of 3 dimensional vector fields (BT-zero singularity) with symmetries. The main technique applied to the computation is the combination of a linear grading function and the method of multiple Lie brackets. We introduce new notations for block matrices to simplify the expression of block matrices. The new notations help to prove the non-degeneracy of huge size matrices by an induction technique.     

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 147–156, July, 2007.