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Register a new userEuler-like vector fields, deformation spaces and manifolds with filtered structure
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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.
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The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.
In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${rm Ric} = 2 F^2$, ${rm Ric}=0$ and ${rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and behaves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the connection and their respective kernels. Some examples are discussed to illustrate the theory.
A connected Riemannian manifold M has constant vector curvature epsilon, denoted by cvc(epsilon), if every tangent vector v in TM lies in a 2-plane with sectional curvature epsilon. By scaling the metric on M, we can always assume that epsilon = -1, 0, or 1. When the sectional curvatures satisfy the additional bound that each sectional curvature is less than or equal to epsilon, or that each sectional curvature is greater than or equal to epsilon, we say that, epsilon, is an extremal curvature. In this paper we study three-manifolds with constant vector curvature. Our main results show that finite volume cvc(epsilon) three-manifolds with extremal curvature epsilon are locally homogenous when epsilon=-1 and admit a local product decomposition when epsilon=0. As an application, we deduce a hyperbolic rank-rigidity theorem.
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.
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