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Register a new userA Note On Conformal Vector Fields Of $(alpha,beta)$-Spaces
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Guojun Yang
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In this paper, we characterize conformal vector fields of any (regular or singular) $(alpha,beta)$-space with some PDEs. Further, we show some properties of conformal vector fields of a class of singular $(alpha,beta)$-spaces satisfying certain geometric conditions.
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In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat $(alpha,beta)$-space of non-Randers type in dimension $nge 3$, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fields are not necessarily homothetic.
In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(alpha,beta)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of $(alpha,beta)$ spaces under certain curvature conditions. Besides, we construct a family of non-homothetic conformal vector fields on a family of locally projectively Randers spaces.
The aim of these notes is to relate covariant stochastic integration in a vector bundle $E$ (as in Norris cite{Norris}) with the usual Stratonovich calculus via the connector $K:TE rightarrow E$ (cf. e.g. Paterson cite{Paterson} or Poor cite{Poor}) which carries the connection dependence.
The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of Euler-like vector fields on V. We shall explain how to approach this bijection through the deformation to the normal cone that is associated to the embedding of M into V. The second purpose is to study generalizations to smooth manifolds equipped with Lie filtrations. Following in the footsteps of several others, we shall define a deformation to the normal cone that is appropriate to this context, and relate it to Euler-like vector fields and tubular neighborhood embeddings.
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
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